Vrije Universiteit Brussel - Faculty of Science and Bio...

Vrije Universiteit Brussel - Faculty of Science and Bio-engineering Sciences

Group representations:

idempotents in group algebras

and applications to units

Graduation thesis submitted in fulfillment of the requirements for the degree of Doctor in Sciences

Author:Inneke Van Gelder

Promotor:Prof. Dr. Eric Jespers

Copromotor:Prof. Dr. Gabriela Olteanu

march 2015

AC K N OW L E D G M E N T S

Ik zou graag iedereen willen bedanken die, elk op zijn manier, heeft bijgedragen

tot de realisatie van deze doctoraatsthesis.

Eerst en vooral zou ik graag mijn promotor Eric Jespers willen bedanken.

Hij heeft zijn enthousiasme voor groepsringen op mij overgedragen door zijn

interessante cursussen en boeiende lezingen. Door zijn ruime kennissenkring

kwam ik in contact met verschillende vriendelijke wiskundigen waar ik veel

van heb kunnen leren.

I owe a very special thank you to my copromotor Gabriela Olteanu. I would

like to thank her for her hospitality during my stays in Romania, for her

support, both professional and personal. Thanks for the nice cooperations

and the very careful proofreading of our joint works.

Also my gratitude is expressed to my co-authors Andreas Bachle, Mauricio

Caicedo, Angel del Rıo, Florian Eisele and Ann Kiefer for the pleasant and

enriching collaborations.

Thanks to Allen Herman and Alexander Konovalov for the very interesting

discussions about GAP and wedderga.

Vervolgens zou ik ook graag mijn collega’s Andreas, Ann, Mauricio en Sara

willen bedanken voor het nalezen van en hun kritische opmerkingen op een

eerste versie van mijn doctoraatsthesis. Graag zou ik alle collega’s van de

Vrije Universiteit Brussel, en in het bijzonder Philippe, Sara, Ann, Karen en

Timmy, willen bedanken voor de fijne lunches, ontspannende koffiepauzes, toffe

babbels, filmavonden, wiskundige nevenactiviteiten en zoveel meer!

Verder ben ik ook dank verschuldigd aan het Fonds voor Wetenschappelijk

Onderzoek om mij gedurende vier jaar financieel te ondersteunen. Hierbij moet

ik ook Stefaan Caenepeel bedanken om mij gedurende een jaar een assistenten-

positie aan te bieden in afwachting van een aanstelling door het FWO.

Mijn familie en vrienden wil ik bedanken om mij onvoorwaardelijk te steunen.

Mijn laatste, maar zeker niet de minste dank gaat uit naar mijn man Giel, om

zijn steun en vertrouwen. Ook een heel dikke dankjewel om dit werk van een

mooie omslag te voorzien.

Inneke Van Gelder

maart 2015

C O N T E N T S

contents i

introduction iii

summary ix

publications xix

samenvatting (summary in dutch) xxi

list of notations xxxiii

1 preliminaries 1

1.1 Fixed point free groups . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Quaternion algebras . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Normal bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Number fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Crossed products . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.6 Group rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.7 Wedderburn-Artin decomposition . . . . . . . . . . . . . . . . . 14

1.8 Z-orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.9 Congruence Subgroup Problem . . . . . . . . . . . . . . . . . . 26

1.10 Finite subgroups of exceptional simple algebras . . . . . . . . . 31

1.11 Cyclotomic units . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.12 Bass units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.13 Bicyclic units . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.14 Central units . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2 wedderburn decomposition and idempotents 41

2.1 The Wedderburn decomposition of FG . . . . . . . . . . . . . . 41

2.2 Primitive idempotents of QG . . . . . . . . . . . . . . . . . . . 55

2.3 Primitive idempotents of FG . . . . . . . . . . . . . . . . . . . 61

2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

i

contents

3 exceptional components 67

3.1 Group algebras with exceptional components of type EC2 . . . 68

3.2 Group algebras with exceptional components of type EC1 . . . 72

3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4 central units 101

4.1 Abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.1.1 A new proof of the Bass-Milnor Theorem . . . . . . . . 103

4.1.2 A virtual basis of Bass units . . . . . . . . . . . . . . . . 107

4.2 Strongly monomial groups . . . . . . . . . . . . . . . . . . . . . 110

4.3 Abelian-by-supersolvable groups . . . . . . . . . . . . . . . . . 113

4.3.1 Generalizing the Jespers-Parmenter-Sehgal Theorem . . 114

4.3.2 Reducing to a basis of products of Bass units . . . . . . 120

4.4 Another class within the strongly monomial groups . . . . . . . 124

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5 applications to units of group rings 135

5.1 A subgroup of finite index in U(Z(Cqm o1 Cpn)) . . . . . . . . . 135

5.2 A method to compute U(ZG) up to commensurability . . . . . 137

5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.3.1 U(ZD+16) up to finite index . . . . . . . . . . . . . . . . 142

5.3.2 U(ZSL(2, 5)) up to commensurability . . . . . . . . . . . 143

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

bibliography 147

index 155

ii

I N T RO D U C T I O N

The notion of a group algebra already appeared in a paper of Arthur Cayley

from 1854. However, only after the influential works of Richard Brauer (1901-

1977) and Emmy Noether (1882-1935) on representation theory, the subject

gained attention because of the correspondence between modules of group alge-

bras and group representations. In 1940, Graham Higman posed the following

question in his Ph.D. thesis, for finite groups G and H:

Does ZG ' ZH imply that G ' H?

This problem is referred to as the (integral) isomorphism problem. It was

anticipated for a long time for this conjecture to be true. In 1987, Klaus W.

Roggenkamp and Leonard L. Scott showed that this indeed is the case if G

is a nilpotent group. It was a surprise when Martin Hertweck gave a counter

example to the isomorphism problem in his Ph.D. thesis in 1998. Nowadays,

it is still an important problem to decide for which classes of groups the con-

jecture does hold. In all these investigations, the unit group U(ZG) of ZGplays a fundamental role. It is essential to consider ZG as a Z-order in the

(semisimple) rational group algebra QG and to have a detailed understanding

of the Wedderburn decomposition of QG.

If one proves the equality of two numbers a and b by showing first that

‘a is less than or equal to b’ and then ‘a is greater than or equal to b’,

it is unfair. One should instead show that they are really equal by

disclosing the inner ground for their equality — Emmy Noether

The Wedderburn-Artin Theorem states that a semisimple ring R is isomor-

phic to a product of finitely many ni-by-ni matrix rings over division rings

Di, for some integers ni. However, in the mindset of Emmy Noether, such a

classification is unfair. One should instead aim to construct an explicit isomor-

phism between R and the product of matrix rings. To do this, a first important

step is to calculate the primitive central idempotents e of R to distinguish the

different matrix rings. Secondly, one needs to construct elements in each com-

ponent Re, which play the role of a complete set of matrix units. In particular,

one has to construct a complete set of orthogonal primitive idempotents.

iii

introduction

A classical method for obtaining the primitive central idempotents of a semi-

simple group algebra FG involves computations using the irreducible charac-

ters of G over an algebraic closure of F . However, the known methods to com-

pute the character table of a finite group are very time consuming. Therefore,

in practical applications, the classical description of primitive central idempo-

tents sometimes is of limited use. One would like a character-free description

that only depends on the lattice of subgroups and the characteristic of the

field, i.e. a description completely within FG. Such a description has been

obtained by Aurora Olivieri, Angel del Rıo and Juan Jacobo Simon in 2004 for

the primitive central idempotents of QG when G is a strongly monomial group,

for example an abelian-by-supersolvable group. This method relies on pairs

of subgroups (H,K) of G satisfying some conditions which can be checked in-

side the rational group algebra QG. Such pairs are called strong Shoda pairs

of G. It turns out that each primitive central idempotent is the sum of the

distinct conjugates of ε(H,K) (corresponding to a natural idempotent in the

rational group algebra Q(H/K)) for a strong Shoda pair (H,K), which we de-

note by e(G,H,K). Furthermore, each simple component in the Wedderburn

decomposition is a matrix ring over a crossed product of the finite abelian

group NG(K)/H over a specific cyclotomic field for some strong Shoda pair

(H,K). In 2007, Osnel Broche and Angel del Rıo transfered those results to

the case of semisimple finite group algebras FG for strongly monomial groups

G. For arbitrary semisimple group algebras FG, it remains an open problem

to give a character-free description of the primitive central idempotents and

the Wedderburn decomposition of FG.

For a rational group algebra QG of a finite nilpotent group G, a complete

set of matrix units of an arbitrary simple component QGe(G,H,K) was given,

in 2012, by Eric Jespers, Gabriela Olteanu and Angel del Rıo. In joined work

with Gabriela Olteanu, we gave a similar result for semisimple finite group

algebras FG of nilpotent groups G. Moreover, examples were given to show

that the method can not be extended to, for example, finite metacyclic groups.

Chapter 1 is a preliminary chapter. In Chapter 2, we first study the primitive

central idempotents and the Wedderburn decomposition of group algebras FG

with F a number field and G a strongly monomial group (Theorem 2.1.6). This

is a generalization of the results of Aurora Olivieri, Angel del Rıo, Juan Jacobo

Simon and Osnel Broche. Next, we focus on a complete set of matrix units

in the Wedderburn components of QG and FG, with F a finite field, for a

class of finite strongly monomial groups containing some metacyclic groups

(Theorems 2.2.1 and 2.3.4).

iv

introduction

I have never done anything ‘useful’. No discovery of mine has made,

or is likely to make, directly or indirectly, for good or ill,

the least difference to the amenity of the world — Godfrey Harold Hardy

Regardless of his sayings, much of the work of Godfrey Harold Hardy (1877-

1947) did find applications in different branches of science, other than mathe-

matics. Hardy was a number theorist and exactly number theory is the elected

area in pure mathematics to have many applications to other areas, such as

coding theory and internet security. In 1974, Donald Knuth formulated this

as follows: “Virtually every theorem in elementary number theory arises in

a natural, motivated way in connection with the problem of making compu-

ters do high-speed numerical calculations”. Finite group algebras and their

Wedderburn decomposition have applications to coding theory as well. Cyclic

codes can be realized as ideals of group algebras of cyclic groups and many

other important codes appear as ideals of group algebras of non-cyclic groups,

see Section 2.3 for references. A concrete realization of the Wedderburn de-

composition also allows applications to many other topics, for example to the

investigation of the group of automorphisms of group rings, as shown by Au-

rora Olivieri, Angel del Rıo and Juan Jacobo Simon in 2006.

In this thesis, we focus on the applications to the group of units of RG, where

R is the ring of integers of a number field F . The main example is the group of

units of integral group rings. Only for very few finite non-commutative groups

G, a presentation of the group U(ZG) is known. However, Carl Ludwig Siegel,

Armand Borel and Harish-Chandra showed, in a much more general setting,

that U(RG) is always finitely generated, if G is finite. Therefore, we are

satisfied with finding finitely many generators of U(RG), and in particular of

U(ZG). If E is a complete collection of primitive central idempotents of FG,

then

RG ⊆⊕e∈E

RGe ⊆⊕e∈E

FGe = FG

and each FGe ' Mne(De) for some integers ne and some division rings De.

Since both RG and⊕

e∈E RGe are Z-orders in FG, we know that U(RG) is

of finite index in⊕

e∈E U(RGe). If we choose an order Oe in each De, then

also GLne(Oe) and U(RGe) have a common subgroup which is of finite index

in both. This means that first, we have to find generating sets of units in

GLne(Oe), which is generated (up to finite index) by SLne(Oe) and the matri-

ces with diagonal entries in U(Z(Oe)). So, the problem reduces to describing

SLne(Oe) and U(Z(Oe)).

v

introduction

In Chapter 3, we classify the finite groups G such that, for a fixed abelian

number field F , for all Wedderburn components Mn(D) in the group algebra

FG, the corresponding SLn(O), for any Z-order O in D, is generated by the

elementary matrices over a two-sided ideal in O (Theorems 3.1.2 and 3.2.21).

The components Mn(D) where this is not possible are the so-called exceptional

components. This investigation is a generalization of a result from Mauricio

Caicedo and Angel del Rıo who dealt with QG. It involves deep results from

Hyman Bass, Bernhard Liehl, Leonid N. Vasersteın and Tyakal Nanjundiah

Venkataramana related to the Congruence Subgroup Problem.

In Chapter 4, we study the central units Z(U(ZG)) for finite groups G. Due

to Hyman Bass and John Willard Milnor (1966) it is well known that, for a

finite abelian group G, the Bass units of the integral group ring ZG generate

a subgroup of finite index in U(ZG). We give a new constructive proof of

this result (Proposition 4.1.1). For non-abelian groups, some constructions of

central units of ZG have been given by Eric Jespers, Guilherme Leal, Michael

M. Parmenter, Sudarshan Sehgal and Raul Antonio Ferraz. This was done

mainly for finite nilpotent groups G. We construct generalized Bass units and

show that they generate a subgroup of finite index in Z(U(ZG)) for finite

strongly monomial groups G (Theorem 4.2.3). For a class within the finite

abelian-by-supersolvable groups G, we can do more and describe a multiplica-

tively independent set (based on Bass units) which generate a subgroup of

finite index in Z(U(ZG)) (Theorem 4.3.8). For another class of finite strongly

monomial groups containing some metacyclic groups, we construct such a set

of multiplicatively independent elements starting from generalized Bass units

(Theorem 4.4.4).

In Chapter 5, we combine the results of the previous chapters to construct

a generating set of U(ZG) up to finite index. This work is a continuation of

a result from Eric Jespers, Gabriela Olteanu and Angel del Rıo from 2012,

that described the unit group of ZG up to finite index for finite nilpotent

groups G. We also continue works of Jurgen Ritter and Sudarshan Sehgal,

and Eric Jespers and Guilherme Leal who showed that under some conditions

the Bass units together with the bicyclic units generate a subgroup of finite

index in U(ZG). If QG does not contain exceptional components, if one can

construct matrix units in each Wedderburn component of QG and moreover, if

one knows a generating set of Z(U(ZG)), then it is possible to describe U(ZG)

up to finite index. We demonstrate this for metacyclic groups Cqm o1Cpn , for

different prime numbers p and q (Theorem 5.1.1). However, if QG has only

exceptional components of type M2(D), then it turns out that SL2(O) can still

vi

introduction

be generated by elementary matrices for a special (i.e. left norm Euclidean) Z-

order O of D (Proposition 5.2.1). This allows us to construct the group of units

of ZG up to finite index for finite groups G, such that QG has only exceptional

components of one type and such that one knows non-central idempotents in

the non-commutative non-exceptional components of QG (Proposition 5.2.2).

Those non-central idempotents are needed to imitate the elementary matrices

with (generalized) bicyclic units in ZG.

vii

S U M M A RY

In this summary, we present our main results.

For the convenience of the reader, Chapter 1 is devoted to a preliminary ex-

position on quaternion algebras, number fields, crossed products, group rings,

Z-orders, cyclotomic units, Bass units and bicyclic units.

In Chapter 2, we give a concrete realization of the Wedderburn decomposi-

tion of group algebras FG of finite strongly monomial groups G over number

fields F . This description is mainly based on the fact that, for rational group

algebras QG of finite strongly monomial groups G, the Wedderburn decompo-

sition is completely described using strong Shoda pairs.

Corollary 2.1.7 [8]

If G is a finite strongly monomial group and F is a number field, then

every primitive central idempotent of FG is of the form eC(G,H,K) for

a strong Shoda pair (H,K) of G and C ∈ CF (H/K). Furthermore, for

every strong Shoda pair (H,K) of G and every C ∈ CF (H/K),

FGeC(G,H,K) 'M[G:E]

(F(ζ[H:K]

)∗στ E/H

),

where E = EF (G,H/K) and σ and τ are defined as follows. Let yK be a

generator of H/K and ψ : E/H → E/K be a left inverse of the projection

E/K → E/H. Then

σgH(ζk) = ζik, if yKψ(gH) = yiK,

τ(gH, g′H) = ζjk, if ψ(gg′H)−1ψ(gH)ψ(g′H) = yjK,

for gH, g′H ∈ E/H and integers i and j.

Next, we obtain more information on the Wedderburn decomposition of

QG and determine a complete set of orthogonal primitive idempotents in each

component determined by a strong Shoda pair provided the twisting τ is trivial.

ix

summary

Theorem 2.2.1 [3]

Let (H,K) be a strong Shoda pair of a finite group G such that the

twisting τ(gH, g′H) = 1 for all g, g′ ∈ NG(K). Let ε = ε(H,K) and

e = e(G,H,K). Let F denote the fixed subfield of QHε under the na-

tural action of NG(K)/H and [NG(K) : H] = n. Let w be a normal

element of QHε/F and B the normal basis determined by w. Let ψ

be the F -isomorphism between QNG(K)ε and the matrix algebra Mn(F )

with respect to the basis B determined as follows:

ψ : QNG(K)ε = QHε ∗NG(K)/H → Mn(F )

xuσ 7→ [x′ ◦ σ]B ,

for x ∈ QHε, σ ∈ Gal(QHε/F ) ' NG(K)/H, where x′ denotes multipli-cation by x on QHε. Let P,A ∈Mn(F ) be defined as follows:

P =

1 1 1 · · · 1 1

1 −1 0 · · · 0 0

1 0 −1 · · · 0 0...

......

. . ....

...

1 0 0 · · · −1 0

1 0 0 · · · 0 −1

and A =

0 0 · · · 0 1

1 0 · · · 0 0

0 1 · · · 0 0...

.... . .

......

0 0 · · · 0 0

0 0 · · · 1 0

.

Then

{xT1εx−1 : x ∈ T2 〈xe〉}

is a complete set of orthogonal primitive idempotents of QGe where we set

xe = ψ−1(PAP−1), T1 is a transversal of H in NG(K) and T2 is a right

transversal of NG(K) in G. By T1 we denote the element 1|T1|

∑t∈T1

t in

QG.

We apply this result in Corollary 2.2.5 to all metacyclic groups of the form

Cqm o1 Cpn , with p and q different prime numbers. We finish the chapter

with a translation of the above theorem to finite semisimple group algebras

(Theorem 2.3.4, [6]).

In Chapter 3, we classify finite groups G and abelian number fields F such

that FG contains an exceptional component in its Wedderburn decomposi-

tion. Hyman Bass (1964), Leonid N. Vasersteın (1973), Bernhard Liehl (1981),

Tyakal Nanjundiah Venkataramana (1994) and Ernst Kleinert (2000) showed

that, under some conditions, the elementary matrices En(I) for all non-zero

ideals I in any order O in a finite dimensional rational division algebra gen-

x

summary

erate a subgroup of finite index in SLn(O). More precisely, if a matrix ring

Mn(D) over a finite dimensional rational division ring D is not of one of the

following forms:

� n = 1 and D is a non-commutative division ring other than a totally

definite quaternion algebra;

� n = 2 and D equals Q, a quadratic imaginary extension of Q, or a totally

definite quaternion algebra with center Q,

then [SLn(O) : En(I)] <∞ for any order O in D and any non-zero ideal I in

O.

A simple finite dimensional rational algebra is called exceptional if it is

in the list above. The exceptional simple algebras occurring as Wedderburn

components of a group algebra, are very restricted.

Corollary 1.9.9 [7]

If a simple finite dimensional rational algebra is an exceptional component

of some group algebra FG for some number field F , then it is of one of

the following types:

EC1: a non-commutative division ring other than a totally definite quater-

nion algebra;

EC2: M2(Q), M2(Q(√−1)), M2(Q(

√−2)), M2(Q(

√−3)), M2

Ä−1,−1

Q

ä,

M2

Ä−1,−3

Q

ä, M2

Ä−2,−5

Q

ä.

We first classify all exceptional components of type EC2 occurring in the

Wedderburn decomposition of group algebras of finite groups over arbitrary

number fields. We do this by giving a full list of finite groups G, number

fields F and exceptional components M2(D) such that M2(D) is a faithful

Wedderburn component of FG.

Theorem 3.1.2 [9]

Let F be a number field, G be a finite group and B a simple exceptional

algebra of type EC2. Then B is a faithful Wedderburn component of FG

if and only if G, F , B is a row listed in Table 2 on page 70.

Secondly, we classify F -critical groups, i.e. groups G such that FG has an

exceptional component of type EC1 in its Wedderburn decomposition, but

xi

summary

no proper quotient has this property. Note that any group H such that FH

has a non-commutative division ring (not totally definite quaternion) in its

Wedderburn decomposition has an epimorphic F -critical image G such that if

an exceptional component D of type EC1 appears as a faithful Wedderburn

component of FG, then also FH has D as a simple component.

Theorem 3.2.21 [9]

Let D be a division ring and F an abelian number field, p and q different

odd prime numbers. Then D is a Wedderburn component of FG for an

F -critical group G if and only if one of the following holds:

(a) D =(−1,−1

F

), G ∈ {SL(2, 3), Q8}, F is totally imaginary and both,

e2(F/Q) and f2(F/Q), are odd;

(b) D =Ä−1,−1F (ζp)

ä, G ∈ {SL(2, 3)×Cp, Q8×Cp}, gcd(p, |G|/p) = 1, op(2)

is odd, F is totally real and both, e2(F (ζp)/Q) and f2(F (ζp)/Q), are

odd;

(c) D =(−1,(ζp−ζ−1

p )2

F (ζp+ζ−1p )

), G = Cp o2 C4, p ≡ −1 mod 4, F totally

imaginary, Q(ζp)∩F ⊆ Q(ζp+ζ−1p ) and both, ep(F/Q) and fp(F/Q),

are odd;

(d) D =(−1,(ζp−ζ−1

p )2

F (ζq,ζp+ζ−1p )

), G = Cq × (Cp o2 C4), p ≡ −1 mod 4, oq(p)

odd, F is totally real and both, ep(F (ζq)/Q) and fp(F (ζq)/Q), are

odd;

(e) D = (K(ζp)/K, σ, ζk) with Schur index nk , G = 〈a〉p ok 〈b〉n with

n ≥ 8, gcd(p, n) = 1, b−1ab = ar, and both k and nk are divisible

by all the primes dividing n. Here K = F (ζk, ζp + ζrp + ...+ ζrnk−1

p )

and σ : F (ζpk)→ F (ζpk) : ζp 7→ ζrp ; ζk 7→ ζk. Moreover Q(ζp) ∩ F ⊆Q(ζp + ζrp + ... + ζr

nk−1

p ) and one of the conditions (i) - (iii) from

Theorem 3.2.20 holds. Furthermore

min

®l ∈ N

∣∣∣∣ pf − 1

gcd(pf − 1, e)≡ 0 mod

k

gcd(k, l)

´=n

k

with e = ep(F (ζpk)/K) and f = fp(K/Q).

xii

summary

Essential here is to use the classification of finite subgroups of division rings

by Shimshon Avraham Amitsur and the classification of maximal finite sub-

groups of 2 × 2-matrices over totally definite quaternion algebras with center

Q given by Gabriele Nebe.

In Chapter 4, we investigate the group of central units of ZG. First, we give

a new constructive proof for the famous Bass-Milnor result avoiding K-theory.

Additionally, we construct a virtual basis in the unit group of ZG for finite

abelian groups G.

Corollary 4.1.6 [4]

Let G be a finite abelian group. For every cyclic subgroup C of G, choose

a generator aC of C and for every k coprime to the order of C, choose an

integer mk,C with kmk,C ≡ 1 mod |C|. Thenßuk,mk,C (aC) : C cyclic subgroup of G, 1 < k <

|C|2, gcd(k, |C|) = 1

™is a virtual basis of U(ZG). Moreover, for any Bass unit uk,m(g) in ZGwe have

uk,m(g)c = h uk0,mk0,C (aC)n0 uk1,mk1,C (aC)n1 ,

for C = 〈g〉, an element h ∈ G and integers c, n0, n1, k0, k1 such that

1 ≤ k0, k1 ≤ |C|2 , g = a±k1C and k0 ≡ ±kk1 mod |C|.

For finite non-abelian groups G, we restrict to strongly monomial groups

because of the detailed description of QG in this case.

Theorem 4.2.3 [5]

Let G be a finite strongly monomial group. The group generated by the

generalized Bass units bnG,H′ , with b = uk,m(1 − H ′ + hH ′) for a strong

Shoda pair (H,K) of G, h ∈ H and nG,H′ the minimal positive integer

such that bnG,H′ ∈ ZG, contains a subgroup of finite index in Z(U(ZG)).

Since we know the rank of Z(U(ZG)), we know a priori the number of

elements in a virtual basis of Z(U(ZG)).

xiii

summary

Theorem 4.2.1 [3]

Let G be a finite strongly monomial group. The rank of Z(U(ZG)) equals

∑(H,K)

Çφ([H : K])

k(H,K)[NG(K) : H]− 1

å,

where (H,K) runs through a complete and non-redundant set of strong

Shoda pairs of G, h is such that H = 〈h,K〉 and

k(H,K) =

ß1 if hhn ∈ K for some n ∈ NG(K);

2 otherwise.

Let u ∈ U(Z 〈g〉), for g ∈ G. Consider a subnormal series

N : N0 = 〈g〉�N1 �N2 � · · ·�Nm = G.

We define cN0 (u) = u and

cNi (u) =∏h∈Ti

cNi−1(u)h,

where Ti is a transversal for Ni in Ni−1, and prove that this construction

behaves well. Define

Sg = {l ∈ U(Z/|g|Z) : g is conjugate with gl in G}

and denote Sg = 〈Sg,−1〉. This construction yields a virtual basis of the group

Z(U(ZG)) in the following setting.

Theorem 4.3.8 [5]

Let G be a finite abelian-by-supersolvable group, such that every cyclic

subgroup of order not a divisor of 4 or 6, is subnormal in G. Let R denote

a set of representatives of Q-classes of G. For g ∈ R, choose a transversal

Tg of Sg in U(Z/|g|Z) containing 1 and for every k ∈ Tg \ {1} choose an

integer mk,g with kmk,g ≡ 1 mod |g|. For every g ∈ R of order not a

divisor of 4 or 6, choose a subnormal series Ng from 〈g〉 to G, which is

normalized by NG(〈g〉). Then{cNg (uk,mk,g (g)) : g ∈ R, k ∈ Tg \ {1}

}is a virtual basis of Z(U(ZG)).

xiv

summary

Next, we focus on another subclass of the finite strongly monomial groups.

Let H be a finite group and K a subgroup of H such that H/K = 〈gK〉 is a

cyclic group of order pn. Let k be a positive integer coprime with p and let r

be an arbitrary integer. For every 0 ≤ s ≤ n, we set

css(H,K, k, r) = 1

and, for 0 ≤ j ≤ s − 1, we construct recursively the following products of

generalized Bass units of ZH:

csj(H,K, k, r) =

Ö ∏h∈〈gpn−j ,K〉

uk,opn (k)nH,K (grpn−s

h“K + 1− “K)

èps−j−1

Ñs−1∏l=j+1

csl (H,K, k, r)−1

é(j−1∏l=0

cs+l−jl (H,K, k, r)−1

).

Theorem 4.4.4 [3]

LetG be a finite strongly monomial group such that there exists a complete

and non-redundant set S of strong Shoda pairs (H,K) of G, with the

property that each [H : K] is a prime power. For every (H,K) ∈ S, let TKbe a right transversal of NG(K) in G, let I(H,K) be a set of representatives

of U(Z/[H : K]Z) modulo 〈NG(K)/H,−1〉 containing 1 and let [H : K] =

pn(H,K)

(H,K) , with p(H,K) a prime number. The set ∏t∈TK

∏x∈NG(K)/H

cn(H,K)

0 (H,K, k, x)t : (H,K) ∈ S, k ∈ I(H,K) \ {1}

is a virtual basis of Z(U(ZG)).

The class of groups mentioned in Theorem 4.4.4, contains the metacyclic

groups Cqm o1 Cpn and we apply our result in Corollary 4.4.5.

In Chapter 5, we first apply Corollaries 2.2.5 and 4.4.5 to construct explicitly

generators for three nilpotent subgroups of U(ZG) that together generate a

subgroup of finite index in U(Z(Cqm o1 Cpn)).

xv

summary

Theorem 5.1.1 [3]

Let p and q be different prime numbers. Let G = Cqm o1 Cpn be a finite

metacyclic group with Cpn = 〈b〉 and Cqm = 〈a〉. Assume that either

q 6= 3, or n 6= 1 or p 6= 2. For every j = 1, . . . ,m, let Kj =¨aqj∂, let

Fj be the center of QGε(〈a〉 ,Kj), fix a normal element wj of Q(ζqj )/Fjand let ψj be the Fj-isomorphism between QGε(〈a〉 ,Kj) and the matrix

algebra Mpn(Fj) with respect to the normal basis Bj associated to wj ,

determined as follows:

ψj : QGε(〈a〉 ,Kj) = Q 〈a〉 ε(〈a〉 ,Kj) ∗G/ 〈a〉 → Mpn(Fj)

xuσ 7→ [x′ ◦ σ]Bj ,

for x ∈ Q 〈a〉 ε(〈a〉 ,Kj), σ ∈ G/ 〈a〉, where x′ denotes multiplication by x

on Q 〈a〉 ε(〈a〉 ,Kj). Let xj = ψj−1(P )bε(〈a〉 ,Kj)ψj

−1(P )−1, with

P =

1 1 1 · · · 1 1

1 −1 0 · · · 0 0

1 0 −1 · · · 0 0...

......

. . ....

...

1 0 0 · · · −1 0

1 0 0 · · · 0 −1

,

and tj be a positive integer such that tjxkj ∈ ZG for all k with 1 ≤ k ≤ pn.

The following two groups are finitely generated nilpotent subgroups of

U(ZG):

V +j =

≠1 + pnt2jyx

hj bx−kj : y ∈fl〈a〉〈b〉, h, k ∈ {1, . . . , pn}, h < k

∑,

V −j =

≠1 + pnt2jyx

hj bx−kj : y ∈fl〈a〉〈b〉, h, k ∈ {1, . . . , pn}, h > k

∑.

Hence V + =∏mj=1 V

+j and V − =

∏mj=1 V

−j are nilpotent subgroups of

U(ZG). Furthermore, the group⟨U, V +, V −

⟩,

with U as in Corollary 4.4.5, is of finite index in U(ZG).

xvi

summary

Next, we generalize results of Jurgen Ritter and Sudarshan Sehgal, and of

Eric Jespers and Guilherme Leal who developed many classes of finite groups in

which U(ZG) is generated up to finite index by the Bass units (denoted B1(G))

and the bicyclic units (denoted B2(G)). The exceptions are the finite groups

G such that their rational group algebra QG has exceptional components or

such that G has non-abelian fixed point free homomorphic images.

Proposition 5.2.2 [7]

Let G be a finite group and let QG =⊕n

i=1 QGei '⊕n

i=1Mni(Di) be

the Wedderburn decomposition of QG. Assume that QG does not contain

exceptional components of type EC1. Also, assume that for each integer

i ∈ {1, . . . , n} such that ni 6= 1 and QGei is not exceptional (of type EC2),

Gei is not fixed point free.

For every exceptional component QGei ' M2(Di), Di has a left norm

Euclidean order Oi. Take a Z-basis Bi of Oi and let ψi : M2(Di)→ QGeibe a Q-algebra isomorphism. For such i, set

Ui :=

ß1 + ψi

Å0 x

0 0

ã, 1 + ψi

Å0 0

x 0

ã: x ∈ Bi

™.

The subgroup U := 〈B1(G) ∪ B2(G) ∪⋃i Ui〉 of QG is commensurable

with U(ZG).

To finish this thesis, we demonstrate our technique on the group D+16 and the

fixed point free group SL(2, 5). As far as we are aware, this is the first technique

known to describe the unit group of ZSL(2, 5) up to commensurability.

xvii

P U B L I C AT I O N S

All results presented in this document have appeared previously (partially,

identically or modified) in the following publications:

[1] Gabriela Olteanu and Inneke Van Gelder. Finite group algebras of nilpo-

tent groups: a complete set of orthogonal primitive idempotents. Finite

Fields Appl., 17(2):157–165, 2011.

[2] Osnel Broche, Allen Herman, Alexander Konovalov, Aurora Olivieri,

Gabriela Olteanu, Angel del Rıo, and Inneke Van Gelder. Wedderga

- Wedderburn Decomposition of Group Algebras. Version 4.5.1+, 2013.

www.cs.st-andrews.ac.uk/ alexk/wedderga.

[3] Eric Jespers, Gabriela Olteanu, Angel del Rıo, and Inneke Van Gelder.

Group rings of finite strongly monomial groups: central units and primi-

tive idempotents. J. Algebra, 387:99–116, 2013.

[4] Eric Jespers, Angel del Rıo, and Inneke Van Gelder. Writing units of

integral group rings of finite abelian groups as a product of Bass units.

Math. Comp., 83(285):461–473, 2014.

[5] Eric Jespers, Gabriela Olteanu, Angel del Rıo, and Inneke Van Gelder.

Central units of integral group rings. Proc. Amer. Math. Soc.,

142(7):2193–2209, 2014.

[6] Gabriela Olteanu and Inneke Van Gelder. Construction of minimal non-

abelian left group codes. Des. Codes Cryptogr., 2014. doi:10.1007/s10623-

014-9922-z.

[7] Florian Eisele, Ann Kiefer, and Inneke Van Gelder. Describing units

of integral group rings up to commensurability. J. Pure Appl. Algebra,

219(7):2901–2916, 2015.

[8] Gabriela Olteanu and Inneke Van Gelder. On idempotents and the number

of simple components of semisimple group algebras. arXiv, abs/1411.5929.

preprint.

xix

http://www.cs.st-andrews.ac.uk/~alexk/wedderga

http://dx.doi.org/10.1007/s10623-014-9922-z

http://dx.doi.org/10.1007/s10623-014-9922-z

http://arxiv.org/abs/1411.5929

publications

[9] Andreas Bachle, Mauricio Caicedo, and Inneke Van Gelder. A classifi-

cation of exceptional components in group algebras over abelian number

fields. arXiv, abs/1412.5458. preprint.

xx


S A M E N VAT T I N G ( S U M M A RY I N D U T C H )

In deze samenvatting schetsen we kort de geschiedenis van de studie van groeps-

ringen en stellen we onze hoofdresultaten voor.

In 1940 stelde Graham Higman de volgende vraag in zijn doctoraatsthesis,

voor twee eindige groepen G en H:

Volgt uit het isomorfisme ZG ' ZH dat G ' H?

Dit probleem is gekend als het (gehele) isomorfismeprobleem. Lange tijd werd

ervan uitgegaan dat deze conjectuur waar was. In 1987 toonden Klaus W.

Roggenkamp en Leonard L. Scott dat deze stelling inderdaad opgaat als G

een eindige nilpotente groep is. Het kwam dan ook als een verrassing toen

Martin Hertweck in 1998 in zijn doctoraatsthesis een tegenvoorbeeld gaf voor

het isomorfismeprobleem. Desalniettemin is het vandaag de dag nog steeds

een belangrijk probleem om te beslissen voor welke klassen van groepen de

conjectuur wel geldig blijft. In dit onderzoek speelt de eenhedengroep U(ZG)

een fundamentele rol. Hierin is het essentieel om ZG te beschouwen als een Z-

order in de (semisimpele) rationale groepsalgebra QG en om een gedetailleerd

inzicht te hebben in de Wedderburndecompositie van QG.

Als men de gelijkheid van twee getallen a en b bewijst door eerst te tonen

dat ‘a kleiner is of gelijk aan b’ en dan ‘a groter is of gelijk aan b’, is dat

oneerlijk. Men zou in plaats daarvan moeten tonen dat ze echt gelijk zijn

door de diepere reden van hun gelijkheid te belichten — Emmy Noether

De Wedderburn-Artin Stelling stelt dat een semisimpele ring R isomorf is

met een product van eindig veel ni × ni matrixringen over (scheve) lichamen

Di, voor zekere natuurlijke getallen ni. In de gedachtegang van Emmy Noether

is zo’n classificatie echter oneerlijk. In plaats daarvan moet men streven naar

de constructie van een expliciet isomorfisme tussen R en het product van ma-

trixringen. Een eerste belangrijke stap hiervoor, is het bepalen van primitieve

centrale idempotenten e van R, die de verschillende matrixringen van elkaar

onderscheiden. Ten tweede moet men elementen in elke component Re con-

strueren die de rol spelen van een volledige verzameling matrixeenheden. In

xxi

samenvatting (summary in dutch)

het bijzonder tracht men een volledige verzameling van orthogonale primitieve

idempotenten te bepalen.

Een klassieke methode voor het bepalen van primitieve centrale idempoten-

ten in een semisimpele groepsalgebra FG brengt berekeningen met zich mee

die gebruik maken van de irreduciebele karakters van G, over een algebraısche

sluiting van F . De gekende methoden om karaktertabellen van eindige groepen

te berekenen zijn echter tijdrovend. Daarom is de klassieke beschrijving van

primitieve centrale idempotenten soms slechts beperkt bruikbaar voor prak-

tische toepassingen. Men verkiest een karaktervrije beschrijving die enkel af-

hangt van de verzameling van deelgroepen en de karakteristiek van het lichaam,

d.w.z. een beschrijving volledig in FG. Zo’n beschrijving werd bekomen door

Aurora Olivieri, Angel del Rıo en Juan Jacobo Simon in 2004 voor de primi-

tieve centrale eenheden van QG, in het geval dat G een sterk monomiale groep

is, bijvoorbeeld abels-bij-superoplosbaar. Deze methode steunt op paren van

deelgroepen (H,K) van G die aan enkele voorwaarden voldoen. Deze paren

noemen we sterke Shoda paren van G. Het blijkt dat elke primitieve cen-

trale idempotent de som is van de verschillende geconjugeerden van ε(H,K)

(een natuurlijk idempotent in de rationale groepsalgebra Q(H/K)), voor een

sterk Shoda paar (H,K). We noteren dit element met e(G,H,K). Elke Wed-

derburncomponent is bovendien een matrixring over een kruisproduct van de

eindige abelse groep NG(K)/H, over een bepaald cyclotomisch lichaam voor

een sterk Shoda paar (H,K). In 2007 vertaalden Osnel Broche en Angel del

Rıo deze resultaten naar het geval van semisimpele eindige groepsalgebra’s

FG voor sterk monomiale groepen G. Voor willekeurige semisimple groepsal-

gebra’s FG blijft het een open probleem om een karaktervrije beschrijving van

de primitieve centrale idempotenten en de Wedderburndecompositie van FG

te geven.

Voor de rationale groepsalgebra QG van een eindige nilpotente groep G be-

schreven Eric Jespers, Gabriela Olteanu en Angel del Rıo in 2012 een volledige

verzameling van matrixeenheden in een willekeurige enkelvoudige component

QGe(G,H,K). In samenwerking met Gabriela Olteanu gaven we een gelijk-

aardig resultaat voor semisimpele eindige groepsalgebra’s FG voor nilpotente

groepen G. Bovendien tonen voorbeelden aan dat de methode niet kan uitge-

breid worden naar, bijvoorbeeld, eindige metacyclische groepen.

In Hoofdstuk 1 geven we een inleiding tot quaternionenalgebra’s, getal-

lenlichamen, kruisproducten, groepsringen, Z-orders, cyclotomische eenheden,

Bass eenheden en bicyclische eenheden.

xxii


In Hoofdstuk 2 geven we een concrete realisatie van de Wedderburndecompo-

sitie van groepsalgebra’s FG van eindige sterk monomiale groepen over getal-

lenlichamen F . Deze beschrijving is hoofdzakelijk gebaseerd op de beschrijving

van Aurora Olivieri, Angel del Rıo, Juan Jacobo Simon en Osnel Broche en het

feit dat voor rationale groepsalgebra’s QG van eindig sterk monomiale groepen

G, de Wedderburndecompositie volledig bepaald is door sterke Shoda paren.

Gevolg 2.1.7 [8]

Zij G een eindige sterk monomiale groep en F een getallenlichaam. Dan

is elke primitieve centrale idempotent van FG van de vorm eC(G,H,K)

voor een sterk Shoda paar (H,K) van G en C ∈ CF (H/K). Bovendien,

voor elk sterk Shoda paar (H,K) van G en elke C ∈ CF (H/K),

FGeC(G,H,K) 'M[G:E]

(F(ζ[H:K]

)∗στ E/H

),

waar E = EF (G,H/K) en σ en τ gedefinieerd zijn als volgt. Zij yK een

voortbrenger voor H/K en ψ : E/H → E/K een links inverse van de

projectie E/K → E/H. Dan

σgH(ζk) = ζik, als yKψ(gH) = yiK,

τ(gH, g′H) = ζjk, als ψ(gg′H)−1ψ(gH)ψ(g′H) = yjK,

voor gH, g′H ∈ E/H en natuurlijke getallen i en j.

Vervolgens verkrijgen we meer informatie over de Wedderburndecompositie

van QG en bepalen we een volledige verzameling van orthogonale primitieve

idempotenten in elke Wedderburncomponent die bepaald wordt door een sterk

Shoda paar met een triviale afbeelding τ .

Stelling 2.2.1 [3]

Zij (H,K) een sterk Shoda paar van een eindige groep G zodat voor

alle g, g′ ∈ NG(K) geldt dat τ(gH, g′H) = 1. Zij ε = ε(H,K) en e =

e(G,H,K). Zij F het deellichaam van QHε dat invariant is onder de actie

van NG(K)/H en stel [NG(K) : H] = n. Zij w een normaal element van

QHε/F en B de normale basis bepaald door w. Zij ψ het F -isomorfisme

tussen QNG(K)ε en de matrixalgebra Mn(F ) ten opzichte van de basis B

bepaald als volgt:

ψ : QNG(K)ε = QHε ∗NG(K)/H →Mn(F ) : xuσ 7→ [x′ ◦ σ]B ,

xxiii


voor x ∈ QHε, σ ∈ Gal(QHε/F ) ' NG(K)/H, waar x′ staat voor devermenigvuldiging met x op QHε. Zij P,A ∈ Mn(F ) gedefinieerd alsvolgt:

P =

1 1 1 · · · 1 1

1 −1 0 · · · 0 0

1 0 −1 · · · 0 0...

......

. . ....

...

1 0 0 · · · −1 0

1 0 0 · · · 0 −1

en A =

0 0 · · · 0 1

1 0 · · · 0 0

0 1 · · · 0 0...

.... . .

......

0 0 · · · 0 0

0 0 · · · 1 0

.

Dan is

{xT1εx−1 : x ∈ T2 〈xe〉}

een volledige verzameling van orthogonale primitieve idempotenten van

QGe met xe = ψ−1(PAP−1), T1 een transversaal van H in NG(K) en T2

een rechts transversaal van NG(K) in G. Met T1 noteren we het element1|T1|

∑t∈T1

t in QG.

We passen onze resultaten toe in Gevolg 2.2.5 op alle metacyclische groepen

van de vorm Cqm o1 Cpn , waarbij p en q verschillende priemgetallen zijn. We

eindigen het hoofdstuk met een vertaling van bovenstaande resultaten naar

semisimpele eindige groepsalgebra’s (Stelling 2.3.4, [6]).

Ik heb nooit iets ‘nuttigs’ gedaan. Geen enkele ontdekking van mij

heeft direct of indirect ook maar de minste bijdrage geleverd,

ten goede of ten kwade, aan de leefbaarheid van de wereld

en zal dat waarschijnlijk ook nooit doen. — Godfrey Harold Hardy

Ongeacht zijn uitspraken vond het werk van Godfrey Harold Hardy (1877-

1947) wel toepassingen in verschillende takken van de wetenschap buiten de

wiskunde. Hardy was een getaltheoreticus en getaltheorie is nu net het uit-

gelezen gebied binnen de zuivere wiskunde om verscheidene toepassingen te

hebben, denk maar aan codetheorie en internetbeveiliging. In 1974 formu-

leerde Donald Knuth dit als volgt: “Zo goed als elke stelling in de elementaire

getaltheorie staat op een natuurlijke manier in verband met het probleem om

computers numerieke berekeningen aan hoge snelheid te laten uitvoeren”. Ook

eindige groepsalgebra’s en hun Wedderburndecompositie hebben toepassingen

binnen de codetheorie. Cyclische codes kunnen gerealiseerd worden als idealen

in groepsalgebra’s van cyclische groepen en ook vele andere belangrijke codes

xxiv


verschijnen als idealen in groepsalgebra’s van niet-cyclische groepen, zie Sectie

2.3 voor referenties. Een concrete realisatie van de Wedderburndecompositie

laat ook vele andere toepassingen toe, bijvoorbeeld het onderzoeken van de

automorfismengroep van groepsringen, zoals aangetoond door Aurora Olivieri,

Angel del Rıo en Juan Jacobo Simon in 2006.

In deze thesis focussen we op de toepassingen voor de eenhedengroep vanRG,

met R de ring van gehele getallen van een getallenlichaam F . Het belangrijkste

voorbeeld is de eenhedengroep van gehele groepsringen. Slechts voor zeer wei-

nig eindige niet-abelse groepenG is een presentatie van de groep U(ZG) gekend.

Nochtans bewezen Carl Ludwig Siegel, Armand Borel en Harish-Chandra, in

een veel algemenere context, wel dat U(RG) altijd eindig voortgebracht is voor

G een eindige groep. Daarom zijn we al tevreden met het vinden van eindig

veel voortbrengers voor U(RG), en in het bijzonder voor U(ZG). Als E een

volledige verzameling van primitieve centrale idempotenten van FG is, dan

RG ⊆⊕e∈E

RGe ⊆⊕e∈E

FGe = FG

en elke FGe ' Mne(De) voor bepaalde natuurlijke getallen ne en (scheve)

lichamen De. Vermits zowel RG als⊕

e∈E RGe een Z-order in FG is, weten

we dat U(RG) van eindige index is in⊕

e∈E U(RGe). Als we een Z-order Oein elke De kiezen, dan hebben GLne(Oe) en U(RGe) een gemeenschappelijke

deelgroep die van eindige index in beide is. Dit betekent dat we in de eerste

plaats een voortbrengende verzameling moeten vinden voor GLne(Oe), die

voortgebracht wordt (op eindige index na) door SLne(Oe) en de matrices met

diagonale elementen in U(Z(Oe)). Het probleem wordt dus herleid tot het

beschrijven van SLne(Oe) en U(Z(Oe)).In Hoofdstuk 3, classificeren we de eindige groepen G waarvoor, gegeven een

willekeurig maar vast abels getallenlichaam F , voor alle Wedderburncomponen-

ten Mn(D) in de groepsalgebra FG, de bijhorende SLn(O), voor elk Z-order

O in D, voortgebracht is door de elementaire matrices over een tweezijdig

ideaal in O. Dit onderzoek is een uitbreiding van een resultaat van Mauricio

Caicedo en Angel del Rıo (2014) en gaat terug tot diepe resultaten van Hy-

man Bass (1964), Leonid N. Vasersteın (1973), Bernhard Liehl (1981), Tyakal

Nanjundiah Venkataramana (1994) en Ernst Kleinert (2000) gerelateerd aan

het congruentiedeelgroepenprobleem. Beter gezegd, als een matrixring Mn(D)

over een eindig dimensionaal rationaal (scheef) lichaam D niet van volgende

vorm is:

xxv


� n = 1 en D is een niet-commutatief scheef lichaam verschillend van een

totaal definiete quaternionenalgebra;

� n = 2 en D is gelijk aan Q, een kwadratische imaginaire uitbreiding van

Q of een totaal definiete quaternionenalgebra met centrum Q,

dan [SLn(O) : En(I)] <∞ voor elk order O in D en elk niet-nul ideaal I van

O.

De componenten Mn(D) van FG die wel voorkomen in de vorige lijst noe-

men we de exceptionele componenten. De exceptionele componenten die kun-

nen optreden als een Wedderburncomponent van een groepsalgebra zijn zeer

beperkt.

Gevolg 1.9.9 [7]

Als een enkelvoudige eindig dimensionale rationale algebra een exceptio-

nele component is van een groepsalgebra FG voor een getallenlichaam F ,

dan is de algebra van een van de volgende types:

EC1: een niet-commutatief scheef lichaam verschillend van een totaal de-

finiete quaternionenalgebra;

EC2: M2(Q), M2(Q(√−1)), M2(Q(

√−2)), M2(Q(

√−3)), M2

Ä−1,−1

Q

ä,

M2

Ä−1,−3

Q

ä, M2

Ä−2,−5

Q

ä.

We classificeren eerst alle exceptionele componenten van type EC2 in de

Wedderburndecompositie van groepsalgebra’s van eindige groepen over wille-

keurige getallenlichamen. Dit doen we door een volledige lijst te geven van

eindige groepen G, getallenlichamen F en exceptionele componenten M2(D)

zodat M2(D) een getrouwe Wedderburncomponent is van FG.

Stelling 3.1.2 [9]

Zij F een getallenlichaam, G een eindige groep en B een enkelvoudige

exceptionele algebra van type EC2. Dan is B een getrouwe Wedderburn-

component van FG als en slechts als G, F en B een rij vormen in Tabel 2

op pagina 70.

Vervolgens classificeren we F -kritische groepen, d.w.z. groepen G zodat

FG een exceptionele component van type EC1 in zijn Wedderburndecompo-

sitie bevat, maar geen enkel echt quotient deze eigenschap heeft. Merk op

dat elke groep H waarvoor FH een niet-commutatief scheef lichaam (geen

xxvi


totaal definiete quaternionenalgebra) in zijn Wedderburndecompositie bevat,

een epimorf F -kritisch beeld G heeft.

Stelling 3.2.21 [9]

Zij D een scheef lichaam, F een abels getallenlichaam en p en q verschil-

lende oneven priemgetallen. Dan is D een Wedderburncomponent van

FG voor een F -kritische groep G als en slechts als een van de volgende

gevallen geldt:

(a) D =(−1,−1

F

), G ∈ {SL(2, 3), Q8}, F is totaal imaginair, e2(F/Q) en

f2(F/Q) zijn oneven;

(b) D =Ä−1,−1F (ζp)

ä, G ∈ {SL(2, 3) × Cp, Q8 × Cp}, ggd(p, |G|/p) = 1,

op(2) is oneven, F is totaal reeel en e2(F (ζp)/Q) en f2(F (ζp)/Q)

zijn oneven;

(c) D =(−1,(ζp−ζ−1

p )2

F (ζp+ζ−1p )

), G = Cp o2 C4, p ≡ −1 mod 4, F is totaal

imaginair, Q(ζp) ∩ F ⊆ Q(ζp + ζ−1p ) en ep(F/Q) en fp(F/Q) zijn

oneven;

(d) D =(−1,(ζp−ζ−1

p )2


), G = Cq × (Cp o2 C4), p ≡ −1 mod 4, oq(p)

oneven, F is totaal reeel en ep(F (ζq)/Q) en fp(F (ζq)/Q) zijn oneven;

(e) D = (K(ζp)/K, σ, ζk) met Schur index nk , G = 〈a〉p ok 〈b〉n waar

n ≥ 8, ggd(p, n) = 1, b−1ab = ar, zowel k als nk zijn deelbaar door

alle priemdelers van n. Hier K = F (ζk, ζp + ζrp + ... + ζrnk−1

p ) en

σ : F (ζpk) → F (ζpk) : ζp 7→ ζrp ; ζk 7→ ζk. Bovendien Q(ζp) ∩ F ⊆Q(ζp + ζrp + ... + ζr

nk−1

p ) en een van de voorwaarden (i) - (iii) uit

Theorem 3.2.20 gelden. Ook

min

®l ∈ N

∣∣∣∣ pf − 1

ggd(pf − 1, e)≡ 0 mod

k

ggd(k, l)

´=n

k

met e = ep(F (ζpk)/K) en f = fp(K/Q).

Hier is het essentieel om gebruik te maken van de classificatie van eindige

deelgroepen van scheve lichamen door Shimshon Avraham Amitsur en de classi-

xxvii


ficatie van maximale eindige deelgroepen in 2×2-matrices over totaal definiete

quaternionenalgebra’s met centrum Q door Gabriele Nebe.

In Hoofdstuk 4, bestuderen we de centrale eenheden Z(U(ZG)) voor eindige

groepen G. Eerst geven we een nieuw en constructief bewijs voor de bekende

stelling van Hyman Bass en John Willard Milnor waarin we het gebruik van

K-theorie vermijden. Bovendien construeren we een virtuele basis in de eenhe-

dengroep van ZG voor eindige abelse groepen G.

Gevolg 4.1.6 [4]

Zij G een eindige abelse groep. Kies voor elke cyclische deelgroep C van

G een voortbrenger aC van C en kies voor elke k relatief priem met de

orde van C een natuurlijk getal mk,C zodat kmk,C ≡ 1 mod |C|. Dan isßuk,mk,C (aC) : C cyclische deelgroep, 1 < k <

|C|2, ggd(k, |C|) = 1

™een virtuele basis van U(ZG). Bovendien geldt voor elke Bass eenheid

uk,m(g) in ZG dat


voor C = 〈g〉, een h ∈ G en gehele getallen c, n0, n1, k0, k1 zodat g = a±k1C ,

1 ≤ k0, k1 ≤ |C|2 en k0 ≡ ±kk1 mod |C|.

Voor sommige niet-abelse groepen zijn constructies van centrale eenheden

van ZG gegeven door Eric Jespers, Guilherme Leal, Michael M. Parmenter, Su-

darshan Sehgal en Raul Antonio Ferraz. Dit werd gedaan vooral voor eindige

nilpotente groepen G. Wij construeren veralgemeende Bass eenheden en to-

nen dat deze een deelgroep voortbrengen die van eindige index is in Z(U(ZG)),

voor eindige sterk monomiale groepen G.

Stelling 4.2.3 [5]

Zij G een eindige sterk monomiale groep. De groep voortgebracht door de

veralgemeende Bass eenheden bnG,H′ , met b = uk,m(1 − H ′ + hH ′) voor

een sterk Shoda paar (H,K) van G, h ∈ H en nG,H′ minimaal zodat

bnG,H′ ∈ ZG, bevat een deelgroep van eindige index in Z(U(ZG)).

Aangezien we de rang van Z(U(ZG)) kennen, weten we op voorhand al exact

hoeveel elementen er in een virtuele basis van Z(U(ZG)) moeten zitten.

xxviii


Stelling 4.2.1 [3]

Zij G een eindige sterk monomiale groep. Dan is de rang van Z(U(ZG))

gelijk aan ∑(H,K)

Çφ([H : K])

k(H,K)[NG(K) : H]− 1

å,

waar (H,K) loopt doorheen een volledige en niet-redundante verzameling

van sterke Shoda paren van G, h is zo dat H = 〈h,K〉 en

k(H,K) =

ß1 als hhn ∈ K voor een n ∈ NG(K);

2 anders.

Zij u ∈ U(Z 〈g〉), met g ∈ G. Beschouw een subnormale rij

N : N0 = 〈g〉�N1 �N2 � · · ·�Nm = G.

We definieren cN0 (u) = u en

cNi (u) =∏h∈Ti

cNi−1(u)h,

met Ti een transversaal voor Ni in Ni−1, en we bewijzen dat deze constructie

zich goed gedraagt. Definieer

Sg = {l ∈ U(Z/|g|Z) : g is geconjugeerd met gl in G}

en noteer Sg = 〈Sg,−1〉. Deze constructie leidt tot een virtuele basis van

Z(U(ZG)) op de volgende manier.

Stelling 4.3.8 [5]

Zij G een eindige abels-bij-superoplosbare groep zodat elke cyclische deel-

groep, van orde niet gelijk aan een deler van 4 of 6, subnormaal is in G.

Zij R een verzameling van representanten van Q-klassen van G. Kies een

transversaal Tg van Sg in U(Z/|g|Z) die 1 bevat voor g ∈ R en kies voor

elke k ∈ Tg \{1} een natuurlijk getal mk,g, zodat kmk,g ≡ 1 mod |g|. Kies

voor elke g ∈ R, van orde niet gelijk aan een deler van 4 of 6, een subnor-

male rij Ng van 〈g〉 naar G, die genormaliseerd wordt door NG(〈g〉). Dan

is {cNg (uk,mk,g (g)) : g ∈ R, k ∈ Tg \ {1}

}een virtuele basis van Z(U(ZG)).

xxix


Vervolgens concentreren we ons op een andere deelklasse van de eindige sterk

monomiale groepen. Zij H een eindige groep en K een deelgroep van H zodat

H/K = 〈gK〉 een cyclische groep is van orde pn. Zij k een natuurlijk getal

relatief priem met p en zij r een willekeurig geheel getal. Voor elke 0 ≤ s ≤ n,

definieren we css(H,K, k, r) = 1 en, voor 0 ≤ j ≤ s−1, construeren we recursief

het volgende product van veralgemeende Bass eenheden van ZH:

csj(H,K, k, r) =

Ö ∏h∈〈gpn−j ,K〉


h“K + 1− “K)

èps−j−1

Ñs−1∏l=j+1

csl (H,K, k, r)−1

é(j−1∏l=0


).

Stelling 4.4.4 [3]

Zij G een eindige sterk monomiale groep zodat er een volledige en niet-

redundante verzameling S van sterke Shoda paren (H,K) van G bestaat

zodat elke [H : K] een macht van een priemgetal is. Voor elke (H,K) ∈ S,

zij TK een rechts transversaal van NG(K) in G, zij I(H,K) een verzameling

representanten van U(Z/[H : K]Z) modulo 〈NG(K)/H,−1〉 die 1 bevat

en zij [H : K] = pn(H,K)

(H,K) , met p(H,K) een priemgetal. Dan is ∏t∈TK

∏x∈NG(K)/H

cn(H,K)

0 (H,K, k, x)t : (H,K) ∈ S, k ∈ I(H,K) \ {1}

een virtuele basis van Z(U(ZG)).

De klasse groepen uit Stelling 4.4.4, bevat de eindige metacyclische groepen

Cqm o1 Cpn en we passen het resultaat toe in Gevolg 4.4.5.

In Hoofdstuk 5 combineren we de resultaten van de voorgaande hoofdstuk-

ken om een voortbrengende verzameling van U(ZG) op eindige index na te

construeren. Dit is een voortzetting van het werk van Eric Jespers, Gabriela

Olteanu en Angel del Rıo uit 2012 waarin ze U(ZG) op eindige index na be-

schreven voor eindige nilpotente groepen. Als QG geen exceptionele compo-

nenten bevat, als men matrixeenheden in elke Wedderburncomponent van QGkan construeren en als men bovendien een voortbrengende verzameling van

xxx


Z(U(ZG)) kent, dan is het mogelijk om U(ZG) te beschrijven op eindige in-

dex na. We demonstreren dit voor metacyclische groepen Cqm o1 Cpn met

verschillende priemgetallen p en q.

Stelling 5.1.1 [3]

Zij p en q verschillende priemgetallen. Zij G = Cqm o1 Cpn een eindige

metacyclische groep waarbij Cpn = 〈b〉 en Cqm = 〈a〉. Onderstel dat ofwel

q 6= 3, ofwel n 6= 1 of p 6= 2. Voor elke j = 1, . . . ,m, stel Kj =¨aqj∂,

stel Fj het centrum van QGε(〈a〉 ,Kj), kies een normaal element wj van

Q(ζqj )/Fj en zij ψj het Fj-isomorfisme tussen QGε(〈a〉 ,Kj) en de ma-

trixalgebra Mpn(Fj) ten opzichte van de normale basis Bj geassocieerd

aan wj , bepaald als volgt:

ψj : QGε(〈a〉 ,Kj) = Q 〈a〉 ε(〈a〉 ,Kj) ∗G/ 〈a〉 → Mpn(Fj)

xuσ 7→ [x′ ◦ σ]Bj ,

met x ∈ Q 〈a〉 ε(〈a〉 ,Kj), σ ∈ G/ 〈a〉, waar x′ staat voor de vermenigvul-

diging met x op Q 〈a〉 ε(〈a〉 ,Kj). Zij xj = ψj−1(P )bε(〈a〉 ,Kj)ψj

−1(P )−1,

met

P =

1 1 1 · · · 1 1

1 −1 0 · · · 0 0

1 0 −1 · · · 0 0...

......

. . ....

...

1 0 0 · · · −1 0

1 0 0 · · · 0 −1

,

en tj natuurlijke getallen zodat tjxkj ∈ ZG voor alle k met 1 ≤ k ≤

pn. Dan zijn de volgende twee groepen eindig voortgebrachte nilpotente

deelgroepen van U(ZG):

V +j =

≠1 + pnt2jyx


∑,

V −j =

≠1 + pnt2jyx


∑.

Bijgevolg zijn V + =∏mj=1 V

+j en V − =

∏mj=1 V

−j nilpotente deelgroepen

van U(ZG). Bovendien is de groep⟨U, V +, V −

⟩,

met U zoals in Gevolg 4.4.5 van eindige index in U(ZG).

xxxi


Wanneer QG toch exceptionele componenten bevat, maar enkel van type

EC2, dan blijkt dat SL2(O) nog steeds voortgebracht kan worden door ele-

mentaire matrices voor een speciaal (links norm Euclidisch) Z-order O van D.

Dit zorgt ervoor dat we de eenhedengroep van ZG op eindige index na kunnen

construeren voor eindige groepen G, zodat QG enkel exceptionele componen-

ten heeft van type EC2 en zodanig dat men niet-centrale idempotenten kent

in de niet-commutatieve niet-exceptionele componenten van QG. Deze niet-

centrale idempotenten zijn nodig om de elementaire matrices te imiteren met

(veralgemeende) bicyclische eenheden in ZG. Dit werk bouwt verder op resulta-

ten van Jurgen Ritter en Sudarshan Sehgal, en van Eric Jespers en Guilherme

Leal die vele klassen van eindige groepen beschreven waar U(ZG) voorgebracht

wordt door de Bass eenheden (genoteerd als B1(G)) en de bicyclische eenheden

(genoteerd als B2(G)) op eindige index na. De uitzonderingen zijn de eindige

groepen G zodat hun rationale groepsalgebra QG exceptionele componenten

bevat of zodat G niet-abelse fixpuntvrije epimorfe beelden heeft.

Stelling 5.2.2 [7]

Zij G een eindige groep en zij QG =⊕n

i=1 QGei '⊕n

i=1Mni(Di) de

Wedderburndecompositie van QG. Onderstel dat QG geen exceptionele

componenten van type EC1 bevat. Stel ook dat voor elke i ∈ {1, . . . , n}waarvoor ni 6= 1 en QGei niet exceptioneel is (van type EC2), Gei niet

fixpuntvrij is.

Het scheef lichaam Di heeft een links norm Euclidisch order Oi voor elke

exceptionele component QGei 'M2(Di). Neem een Z-basis Bi van Oi en

zij ψi : M2(Di)→ QGei een Q-algebra-isomorfisme. Voor zo’n i, definieer

Ui :=

ß1 + ψi

Å0 x

0 0

ã, 1 + ψi

Å0 0

x 0

ã: x ∈ Bi

™.

De deelgroep U := 〈B1(G) ∪ B2(G) ∪⋃i Ui〉 van QG en U(ZG) bevatten

een gezamenlijke deelgroep van eindige index in beiden.

Om de thesis te besluiten, demonstreren we onze techniek op de groep D+16

en de fixpuntvrije groep SL(2, 5). Voor zover we weten is dit de eerste gekende

techniek om de eenhedengroep van ZSL(2, 5) te beschrijven.

xxxii

L I S T O F N O TAT I O N S

[G,H] commutator of G and H, page 1

〈g〉n cyclic group generated by g of order n, page 1

|g| order of a group element g, page 1“H group ring element |H|−1∑h∈H h, page 2

g ”〈g〉, page 2

Fp completion of F with respect to p, page 5›xY orbit sum, page 136

(L/F, τ) classical crossed product, page 10Äa,bF

äquaternion algebra over F , page 3

B1(G) group generated by the Bass units of ZG, page 35

B2(G) group generated by the bicyclic units of ZG, page 36

Cn cyclic group order n, page 1

CenG(α) centralizer of α in G, page 2

CF (G) set of orbits of the faithful characters of G under the

action of Gal(F (ζ|G|)/F ), page 41

EF (G,H/K) stabilizer of any C ∈ CF (H/K) under the action of

NG(H) ∩NG(K), page 44

En(R) group generated by elementary matrices, page 27

e(G,H,K)∑t∈G/CenG(ε(H,K)) ε(H,K)t, page 16

eC(G,H,K)∑t∈G/CenG(εC(H,K)) εC(H,K)t, page 42

ep(L/F ) ramification index, page 7

xxxiii

list of notations

F (χ) field of character values, page 16

FG fixed subfield of F under G, page 2

fp(L/F ) residue degree, page 7

G′ commutator subgroup of G, page 1

Gom H semidirect product of H acting on G with kernel of

order m, page 1

GB2(G) group generated by the generalized bicyclic units of

ZG, page 37

GLn(R) general linear group, page 26

gcd(r,m) greatest common divisor of r and m, page 1

H < G H is a proper subgroup of G, page 1

H ≤ G H is a subgroup of G, page 1

H �G H is a proper normal subgroup of G, page 1

H �G H is a normal subgroup of G, page 1

ind(A) Schur index, page 4

Mn(R) n× n-matrices, page 4

mp(A) p-local index, page 7

NG(H) normalizer of H in G, page 1

nrdA/K(a) reduced norm, page 26

om(r) multiplicative order of r modulo m, page 1

RG group ring, page 12

R ∗ατ G crossed product, page 10

S ⊂ T S is a proper subset of T , page 1

S ⊆ T S is a subset of T , page 1

xxxiv

list of notations

SLn(R) special linear group, page 26

SLn(R,m) congruence subgroup of level m, page 26

uk,mnG,M (1− M + gM) generalized Bass unit, page 111

uk,m(g) Bass unit, page 34

U(R) unit group of R, page 25

vp(m) maximum non-negative integer k such that pk divides

m, page 1

Z(G) center of G, page 1

βei,h generalized bicyclic unit, page 36

βg,h bicyclic unit, page 36

χG induced character, page 17

ε(H,H) “H, page 16

ε(H,K)∏M/K∈M(H/K)(

“K − M), page 16

εC(H,K) |H|−1∑h∈H

∑ψ∈C ψ(hK)h−1, page 42

ηk(ζjn) cyclotomic unit, page 34

γei,h generalized bicyclic unit, page 37

γg,h bicyclic unit, page 36

φ Euler’s totient function, page 8

ζm complex primitive m-th root of unity, page 1

xxxv

1P R E L I M I N A R I E S

One denotes the set of positive integers (without zero) by N. All integers are

denoted by Z and the rational numbers by Q. Further, one refers to the real

numbers by R and the complex numbers by C. For the integers modulo k, one

uses the notation Z/kZ. For integers r and m, one uses gcd(r,m) to refer to

the greatest common divisor of r and m. For integers r,m and p, with p a

prime number and gcd(r,m) = 1, one defines

vp(m) = maximum non-negative integer k such that pk divides m;

om(r) = multiplicative order of r modulo m;

ζm = complex primitive m-th root of unity.

Throughout this thesis, G will be a finite group and F a field. One denotes

by char(F ) the characteristic of F and by FG the group algebra of G over F .

For two sets S and T , one writes S ⊆ T (resp. S ⊂ T ) if S is a subset (resp.

proper subset) of T . The notation H ≤ G (resp. H < G, H � G, H � G)

signifies thatH is a subgroup (resp. proper subgroup, normal subgroup, proper

normal subgroup) of G. For a group element g, one uses the notation |g| for its

order. By Z(G), one denotes the center of G. When g is an element of a group

G with order n, one writes 〈g〉 or 〈g〉n to denote the cyclic group (of order n)

generated by g. If no explicit generator is given, one writes Cn to refer to the

cyclic group of order n. For subgroups G and H of a common group, one writes

[G,H] to denote the commutator of G and H, i.e. the subgroup generated by

all elements g−1h−1gh with g ∈ G and h ∈ H. For [G,G], i.e. the commutator

subgroup of G, one also uses the notation G′. By G om H one denotes a

semidirect product of H acting on G with kernel of order m. For H ≤ G,

g ∈ G and h ∈ H, one writes Hg = g−1Hg and hg = g−1hg. Analogously, for

α ∈ FG and g ∈ G, αg = g−1αg. For H ≤ G, NG(H) denotes the normalizer

1

preliminaries

of H in G. When the characteristic of F does not divide the order of G, one

sets “H = |H|−1∑h∈H h, an idempotent of FG; if H = 〈g〉, then one simply

writes g for ”〈g〉. For each α ∈ FG, CenG(α) denotes the centralizer of α in G.

Furthermore, if F is a field and G a group of automorphisms of F , one writes

FG for the fixed subfield of F under G.

In this work, the algebra FG, the ring RG, for the ring of integers R of a

number field F , and in particular its unit group U(RG) play a prominent role.

A fundamental problem is to find generators and determine the structure of

U(RG), and in particular of U(ZG). This problem has already been widely

studied by Bass, Sehgal, Ritter, Zassenhaus, Higman, Passman, Marciniak,

Hoechsmann, Jespers, del Rıo, Kimmerle, Parmenter and many others. The

study of these objects combines methods from representation theory, K-theory,

number theory and group theory. A complete review on the current state of

the research of group rings and their unit groups is written by Eric Jespers

and Angel del Rıo [JdR]. In this chapter, for the convenience of the reader,

we give a survey on the tools used in the investigation of U(RG).

1.1 fixed point free groups

A finite group G is said to be fixed point free if it has an (irreducible) complex

representation ρ such that 1 is not an eigenvalue of ρ(g), for all 1 6= g ∈ G. A

finite group G is said to be a Frobenius group if it contains a proper non-trivial

subgroup H such that H ∩Hg = {1}, for all g ∈ G \H. The group H is called

a Frobenius complement in G.

Fixed point free groups are exactly the Frobenius complements, as explained

in the book of Donald S. Passman [Pas68, Theorem 18.1.v].

Theorem 1.1.1

A finite group G is fixed point free if and only if G is a Frobenius comple-

ment (in some other finite group).

Hans Zassenhaus described all Frobenius complements and in particular

proved the following result. A proof can be found in [Pas68, Theorem 18.6].

Theorem 1.1.2 (Zassenhaus)

SL(2, 5) is the smallest non-solvable Frobenius complement.

2

1.2 quaternion algebras

1.2 quaternion algebras

Given a field F , as well as two elements a, b ∈ F , the quaternion algebraÄa,bF

äis: Å

a, b

F

ã=

F 〈i, j〉(i2 = a, j2 = b, ij = −ji)

,

where F 〈i, j〉 denotes the free F -algebra on the non-commuting free variables

i and j.

The following proposition determines when a quaternion algebra is a division

ring. A proof can be found in a book by Richard S. Pierce [Pie82, Proposition

1.6].

Proposition 1.2.1

Let F be a field with char(F ) 6= 2. The quaternion algebraÄa,bF

äis a

division ring if and only if the equation ax2 + by2 = z2 does not have a

non-zero solution (x, y, z) in F 3.

A quaternion algebraÄa,bF

äis said to be totally definite if F is contained in

C, is totally real and a and b are totally negative.

1.3 normal bases

Let K be a finite Galois extension of a field F . Often, one is interested in an

F -basis of K. The Normal Basis Theorem answers this need. It can be found

in many standard references, see for example [Art44, Theorem 28].

Theorem 1.3.1 (Normal Basis Theorem)

Let K be a finite normal extension of a field F and let σ1, . . . , σn be the

elements in the Galois group of K over F . There always exists an element

w ∈ K such that the elements σ1(w), . . . , σn(w) are linearly independent

with respect to F .

If K/F is a finite Galois extension, then there exists an element w ∈ K

such that {σ(w) : σ ∈ Gal(K/F )} is an F -basis of K, a so-called normal basis,

whence w is called normal in K/F .

3

preliminaries

1.4 number fields

We recall some basic background on number fields and local Schur indices,

mainly based on the books of Edwin Weiss [Wei98] and Irving Reiner [Rei75].

For a positive integer n and a ring R, one denotes by Mn(R) the n × n-

matrices over R. A central simple algebra over a field F is a finite dimensional

associative algebra A, which is simple, and for which the center is exactly F .

The dimension of a central simple algebra A as a vector space over its center

is always a square; the degree of A is the square root of this dimension. A

field F is called a number field if F is a finite extension of Q. Let F be a

number field and A a finite dimensional central simple F -algebra. A famous

result of Joseph Wedderburn states that A = Mn(D) for some division algebra

D with center F (see for example [Rei75, Theorem 7.4]). The Schur index of

A, denoted by ind(A), is the degree of D. Clearly, A is itself a division algebra

if and only if ind(A) equals the degree of A.

A valuation of a field F is a function ϕ from F into the non-negative reals

such that for all a, b ∈ F :

1. ϕ(a) = 0 if and only if a = 0;

2. ϕ(ab) = ϕ(a)ϕ(b);

3. ϕ(a+ b) ≤ ϕ(a) + ϕ(b).

If the valuation also satisfies the stronger condition ϕ(a+b) ≤ max{ϕ(a), ϕ(b)},then ϕ is said to be non-archimedean. Otherwise, one says that ϕ is archime-

dean.

A valuation ϕ of a field F is always associated to a topology Tϕ on F . For

each a ∈ F , a fundamental system of neighborhoods of a is given by the set of

all

U(a, ε) = {b ∈ F : ϕ(a− b) < ε},

for ε > 0,

Two valuations ϕ and ψ are equivalent if for all a ∈ F , ϕ(a) ≤ 1 if and only

if ψ(a) ≤ 1. Equivalent valuations determine the same topology.

From now on we assume F to be a number field. One way of obtaining

archimedean valuations is the following. The ordinary absolute value | · | is

an archimedean valuation on C. Let σ be an embedding of F in C, then

|σ(a)|, a ∈ F , defines an archimedean valuation on F . In particular, if F has r

4

1.4 number fields

embeddings in R and s pairs of complex embeddings in C, then one can obtain

in this way r + s archimedean valuations.

Now we consider some non-archimedean valuations. Let R be the ring of

integers of F and p a non-zero prime ideal of R. For each non-zero a ∈ F , the

principal ideal aR factors into a product of powers of prime ideals. Let ep(a)

denote the exponent to which p occurs in this factorization. If p does not occur,

we set ep(a) = 0. Also, ep(0) = +∞. Fix 0 < ε < 1 and set vp(0) = 0 and

vp(a) = εep(a) for a ∈ F \ {0}. In this case, vp is a non-archimedean valuation

of F , called the p-adic valuation on F .

The “Big” Ostrowski Theorem states that every archimedean valuation of

F is equivalent to exactly one of the r+ s valuations arising from embeddings

into C and that any non-archimedean valuation of F is equivalent to a p-adic

valuation for a unique prime ideal p in the ring of integers of F . The “Little”

Ostrowski Theorem deals with the case F = Q and is well known (see for

example [Pie82, Theorem 17.3]). The general case is folklore and explained

nicely by Keith Conrad in one of his many handouts [Con].

A prime of F is an equivalence class of valuations of F . One excludes the

trivial valuation ϕ defined by ϕ(0) = 0, ϕ(a) = 1 for a ∈ F \ {0}. Since F is

a number field, there are the infinite primes of F arising from embeddings of

F into C and the finite primes of F , arising from p-adic valuations of F , with

p ranging over the distinct prime ideals in the ring of integers of F . In many

references the primes of F are also called places.

Let ϕ be a valuation of F . A sequence (an)n∈N, an ∈ F , is called a ϕ-

Cauchy sequence if am−an converge to 0 in the topology Tϕ when both m and

n converges to +∞. One says that F is complete if every ϕ-Cauchy sequence

converges. By definition, ‹F (or (‹F , ϕ)) is the completion of F with respect to

ϕ if F is dense in ‹F , the valuation ϕ extends to a valuation ϕ on ‹F and (‹F , ϕ)

is complete.

If ϕ is an archimedean valuation, then the completion of F is either R or Cand in each case ϕ is equivalent to the ordinary absolute value. The completion

of F , with respect to the p-adic valuation vp on F , will be denoted by Fp. One

defines the local Schur index (or local index ) of a central simple F -algebra A

at p as mp(A) = ind(Fp ⊗F A).

Example 1.4.1 If F = Q, then its ring of integers is Z and the non-zero prime

ideals of Z are exactly the ideals pZ with p ranging over the prime numbers.

Hence the finite primes of Q correspond to the prime numbers in N. The

completion of Q at a prime p is the set Qp of p-adic numbers. Furthermore,

5

preliminaries

an infinite prime of Q corresponds to the unique embedding of Q in R and is

often simply denoted by ∞.

The following theorem (see [Rei75, Theorems 31.8 and 32.11]) states that

division rings with a fixed center are uniquely determined up to isomorphism

by reduced fractions modulo Z.

Theorem 1.4.2

Let F be a number field and denote by Pl(F ) the set of all primes (finite

and infinite) of F . There exists an injective map from the isomorphism

classes of division rings with center F to the sequences of reduced fractions

modulo Z in (Q/Z)(Pl(F )). Moreover, this map is defined such that, if

D 7→Årpmp

ãp∈Pl(F )

,

then the denominators equal the local Schur indices mp = mp(D) at the

primes p ∈ Pl(F ).

For a fixed prime p of F , the reduced fractionrpmp

, associated to a division

ring D by the injective map from Theorem 1.4.2, is known as the local Hasse

invariant at p of D.

The following theorem is a well known consequence of the Brauer-Hasse-

Noether-Albert Theorem.

Theorem 1.4.3

Let F be a number field and denote by Pl(F ) the set of all primes (finite

and infinite) of F . Let A be a central simple F -algebra having local Schur

indices (mp(A))p∈Pl(F ). Then ind(A) = lcm(mp(A) : p ∈ Pl(F )).

If a local Schur index equals 1, then the only possible local Hasse invariant

is 11 . If a local Schur index is 2, then the only possible local Hasse invariant

is 12 . Therefore, for a fixed number field F , division rings with center F and

Schur index at most 2, are uniquely determined up to isomorphism by their

local Schur indices. This is no longer true for division rings with bigger Schur

index.

Let L be a number field containing F and let A be a central simple F -algebra.

Here, it is clear that ind(L⊗F A) divides ind(A). One says that L splits A if

and only if L ⊗F A is a matrix ring over L, if and only if ind(L ⊗F A) = 1.

6

1.4 number fields

Remark that if A is a division algebra of degree a prime number, then L does

not split A if and only if L⊗F A is a division algebra.

Let L/F be number fields, R be the ring of integers of F , S the ring of

integers of L and P a finite prime of S. One finds that P ∩R = p, a prime in

R, and one says that P lies over p. The primes lying over a given prime p are

the P1, . . . ,Ps which occur in the prime decomposition of pS = Pe11 · · ·Pes

s .

One also says that Pi divides p. The exponents ei are called the ramification

indices and are denoted by ePip (L/F ). If P is a prime lying over p in the

extension L/F , then R/p can be viewed as a subfield of S/P and one calls

the degree of S/P over R/p the residue degree of P over p and denote it by

fPp (L/F ). If L/F is a normal extension and p is a prime of R, then the Galois

group Gal(L/F ) permutes the primes lying over p transitively. Therefore,

both the ramification index and residue degree are independent on the choice

of primes lying over p and one writes ep(L/F ) and fp(L/F ).

One also knows the following multiplicative rules for normal extensions, see

[Wei98, Proposition 1.6.2].

Theorem 1.4.4

Let K/L/F be normal extensions of number fields. Let p be a prime in

F and P a prime in L lying over p. Then

1. ep(K/F ) = eP(K/L)ep(L/F );

2. fp(K/F ) = fP(K/L)fp(L/F ).

Because of these rules, the notations ep(L/F ) and fp(L/F ) are unambiguous

for normal extensions L/F/Q and rational primes p.

Similar as for ramification index and residue degree, one can also restrict

the computation of local Schur indices to rational primes when working over

abelian number fields. The following theorem is due to Mark Benard [Ben72].

Theorem 1.4.5 (Benard)

Let F be an abelian number field and A a central simple F -algebra. As p

runs over the set of primes lying over the same (infinite or finite) rational

prime p, the local indices mp(A) are all equal to the same positive integer.

One calls the integer defined by the previous theorem the p-local index of A

and denote it by mp(A).

7

preliminaries

The following proposition follows from [Wei98, Proposition 1.7.5 and Theo-

rem 2.3.2].

Proposition 1.4.6

Let K/F be a normal extension of number fields, p a prime number and

let Fp and ›Kp be completions of respectively F and K with respect to p.

Then ep(›Kp/Fp) = ep(K/F ) and fp(›Kp/Fp) = fp(K/F ).

Furthermore, ep(›Kp/Fp)fp(›Kp/Fp) = [›Kp : Fp].

The next proposition follows from [Wei98, Corollary 2.5.9].

Proposition 1.4.7

Let F be a finite normal extension of Q and R be the ring of integers of

F . Let p be a prime number. If r is the number of prime ideals in which

pR splits, then ep(F/Q)fp(F/Q)r = [F : Q].

The following theorem can be used to decide if some field splits a division

ring and can be found in [Rei75, Corollary 31.10] and [Deu68, Satz 2 on p.

118].

Theorem 1.4.8

Let K be an abelian number field extending the number field F . Let D

be a division ring with center F and let p be a rational prime. Then

mp(K ⊗F D) = 1 if and only if mp(D) | [›Kp : Fp].

We conclude this section with some computations and easy examples that

we need later. One denotes by φ Euler’s totient function. For cyclotomic

fields, the ramification index and residue degree is easy to compute, as shown

in [Wei98, Theorems 7.2.4 and 7.4.3].

Theorem 1.4.9

Let Q(ζm) be a cyclotomic field and p a prime number. Write m = psm′

with gcd(p,m′) = 1. One calculates that ep(Q(ζm)/Q) = φ(ps) and

fp(Q(ζm)/Q) = om′(p).

Example 1.4.10 An easy computation shows that e2(Q(√

2)/Q) = 2. By

Proposition 1.4.7, it now automatically follows that f2(Q(√

2)/Q) = 1.

8

1.4 number fields

Example 1.4.11 Consider the following fields for a prime number p 6= 2:

Q(ζp)

Q(ζp + ζ−1p )

Q

Since

ep(Q(ζp)/Q) = p− 1

= [Q(ζp) : Q]

= [Q(ζp) : Q(ζp + ζ−1p )][Q(ζp + ζ−1

p ) : Q]

and

ep(Q(ζp)/Q) = ep(Q(ζp)/Q(ζp + ζ−1p ))ep(Q(ζp + ζ−1

p )/Q),

it follows that

ep(Q(ζp)/Q(ζp + ζ−1p )) = [Q(ζp) : Q(ζp + ζ−1

p )] = 2

and

ep(Q(ζp + ζ−1p )/Q) = [Q(ζp + ζ−1

p ) : Q] =p− 1

2.

Therefore, it immediately follows that fp(Q(ζp)/Q) = fp(Q(ζp)/Q(ζp+ζ−1p )) =

fp(Q(ζp + ζ−1p )/Q) = 1.

The following example can be found in [Lam05, Theorems 6.2.5 and 6.2.24].

Example 1.4.12 Let A =Ä−1,−1

Q

ä. Then R⊗QA =

(−1,−1R)

has Schur index

2, so m∞(A) = 2. One can also show that m2(A) = 2 and mp(A) = 1 for all

odd prime numbers p.

Fixed subfields under cyclic groups are easy to compute in extensions of Qby a primitive root of unity of prime order, see [DF04, Example 14.5.2] for a

proof.

Lemma 1.4.13

Let p be a prime number and r be an integer coprime to p. Let G denote

the cyclic group generated by the automorphism Q(ζp)→ Q(ζp) : ζp 7→ ζrp .

Then Q(ζp)G = Q(ζp + ζrp + . . .+ ζr

op(r)−1

p ).

9

preliminaries

1.5 crossed products

We recall some basic facts about crossed products, based on the books of

Donald S. Passman [Pas89] and Irving Reiner [Rei75].

If R is an associative ring and G is a group, then R∗ατ G denotes the crossed

product with action α : G→ Aut(R) and twisting τ : G×G→ U(R), i.e. R∗ατGis the associative ring

⊕g∈GRug with multiplication given by the following

rules: uga = αg(a)ug and uguh = τ(g, h)ugh, for a ∈ R and g, h ∈ G. Recall

that a classical crossed product , notated by (L/F, τ), is a crossed product

L ∗ατ G, where L/F is a finite Galois extension, G = Gal(L/F ) is the Galois

group of L/F and α is the natural action of G on L. Let F be a field and ζ

be a root of unity in an extension of F . If Gal(F (ζ)/F ) = 〈σn〉 is cyclic of

order n, one can consider the cyclic cyclotomic algebra F (ζ) ∗ατ Gal(F (ζ)/F ),

where α is the natural action on F (ζ), i.e. ασmn = σmn . Also, τ(g, h) is a root

of unity for every g and h in Gal(F (ζ)/F ) and τ is completely determined by

unσn = ζc. One denotes this cyclic cyclotomic algebra by (F (ζ)/F, σn, ζc).

A classical crossed product (L/F, τ) is always a central simple F -algebra,

see [Rei75, Theorem 29.6] for a proof.

Theorem 1.5.1

Let L/F be a finite Galois extension and G = Gal(L/F ) its Galois group.

For any twisting τ : G×G→ U(L), the classical crossed product (L/F, τ)

is a central simple F -algebra.

If the twisting τ is cohomologically trivial, then the classical crossed product

is isomorphic to a matrix algebra over its center. Denote the matrix associated

to an endomorphism f in a basis B as [f ]B . The following theorem from [Rei75,

Corollary 29.8] constructs an explicit isomorphism when τ = 1.

Theorem 1.5.2

Let L/F be a finite Galois extension and n = [L : F ]. The classical crossed

product (L/F, 1) is isomorphic (as F -algebra) to Mn(F ). Moreover, an

isomorphism is given by

ψ : (L/F, 1) → EndF (L) → Mn(F )

xuσ 7→ x′ ◦ σ 7→ [x′ ◦ σ]B ,

for x ∈ L, σ ∈ Gal(L/F ), B an F -basis of L and where x′ denotes

multiplication by x on L.

10

1.6 group rings

The following theorem is due to Gabriela Olteanu [Olt09, Proposition 2.13].

Theorem 1.5.3 (Olteanu)

Let A = (F (ζn)/F, σ, ζm) for F a number field. If p is a prime of F , then

mp(A) divides m. If mp(A) 6= 1 for a finite prime p, then p divides n in

F .

Allen Herman gave a nice review on the computation of local indices for

cyclic cyclotomic algebras in [Her] and he implemented this into the GAP-

package wedderga [BHK+14].

Lemma 1.5.4 (Herman)

A cyclic cyclotomic algebra A = (F (ζn)/F, σ, ζcn) over an abelian number

field F has local index 2 at an infinite prime if and only if F ⊆ R, n > 2

and ζcn = −1.

The following lemma originates from Gerald Janusz [Jan75, Lemma 3.1].

Lemma 1.5.5 (Janusz )

Let p be an odd prime number and F an abelian number field. Let

e = ep(F (ζn)/F ) and f = fp(F/Q). Then

mp(F (ζn)/F, σ, ζcn)

= min

®l ∈ N :

pf − 1


n

gcd(n, cl)

´.

1.6 group rings

We recall some notions from the commendable review book of Cesar Polcino

Milies and Sudarshan K. Sehgal [PMS02].

Let G be a group (not necessarily finite) and R a unitary ring. One denotes

by RG the set of all formal linear combinations of the form

α =∑g∈G

rgg,

where rg ∈ R and only a finite number of coefficients rg are different from 0.

Given two elements α =∑g∈G rgg and β =

∑g∈G sgg ∈ RG, one has that

11

preliminaries

α = β if and only if rg = sg for every g ∈ G. One turns the set RG into a

ring via the following operations. The sum of two elements of RG is defined

component-wise:Ñ∑g∈G

rgg

é+

Ñ∑g∈G

sgg

é=∑g∈G

(rg + sg)g.

Their product is defined byÑ∑g∈G

rgg

é·

Ñ∑g∈G

sgg

é=∑g,h∈G

(rgsh)gh.

It is easy to verify that indeed, with the operations given, RG is a ring with

unity.

One can also define a product of elements in RG with elements λ ∈ R by

λ ·∑g∈G

rgg =∑g∈G

(λrg)g.

This way, RG turns into an R-module and when R is commutative, RG is even

an algebra over R.

The set RG, with the operations as above, is called the group ring of G over

R. In the case where R is commutative, RG is also called the group algebra of

G over R.

When R = Z, one calls ZG the integral group ring, when R = Q, one calls

QG the rational group algebra and when R = F is a finite field and G is a

finite group, one says that FG is a finite group algebra.

Given an element α =∑g∈G rgg ∈ RG, one defines the support of α to be

the set of elements that effectively appear in the expression of α, i.e.

supp(α) = {g ∈ G : rg 6= 0}.

Let R be a ring, G a finite group and N a normal subgroup of G such that

|N | is invertible in R. It is easy to see that the map G→ G“N given by g 7→ g“Nis a group epimorphism with kernel N . Hence, the R-linear extension yields

an isomorphism

R(G/N) ' RG“N. (1)

12

1.6 group rings

An R-module M is called semisimple if every submodule of M is a direct

summand. A ring R is called semisimple if the regular module RR is semi-

simple. Semisimple rings have a clear structure as shown in [PMS02, Theorem

2.5.7].

Theorem 1.6.1

Let R be a ring. Then R is semisimple if and only if R is a direct sum of

a finite number of minimal left ideals.

Heinrich Maschke provided necessary and sufficient conditions on R and G

for the group ring RG to be semisimple, see [PMS02, Theorem 3.4.7].

Theorem 1.6.2 (Maschke)

Let G be a group and R a ring. Then, the group ring RG is semisimple if

and only if the following conditions hold:

1. R is a semisimple ring;

2. G is finite;

3. |G| is invertible in R.

Since fields are always semisimple, Maschke’s Theorem is even easier to state

in the case when R is a field.

Corollary 1.6.3

Let G be a finite group and let F be a field. Then, FG is semisimple if

and only if the characteristic of F does not divide |G|.

The structure of semisimple abelian group rings is well studied by Sam Perlis

and Gordon Loftis Walker in [PW50].

Theorem 1.6.4 (Perlis-Walker)

Let G be a finite abelian group of order n and let F be a field such that

the characteristic of F does not divide n. Then

FG '⊕d|n

kdF (ζd),

where kd = nd[F (ζd):F ] and nd is the number of elements of order d in G.

For F = Q, one gets the following corollary.

13

preliminaries

Corollary 1.6.5

Let G be a finite abelian group of order n. Then

QG '⊕d|n

kdQ(ζd),

where kd is the number of cyclic subgroups of order d of G (or equivalently,

kd is the number of cyclic factors of G of order d).

1.7 wedderburn-artin decomposition

An element e in a ring R is called an idempotent if e2 = e. Clearly, 0 and 1

are idempotents. An idempotent different from these is said to be non-trivial.

For a unitary ring R, a matrix unit in Mn(R) is a matrix whose entries are

all 0 except in one cell, where it is 1. Clearly matrix units with a 1 on the

diagonal, are idempotents in Mn(R).

When R is a semisimple ring, the left ideals of R are determined by idem-

potents as shown in [PMS02, Theorem 2.5.10].

Theorem 1.7.1

Let R be a ring. Then R is semisimple if and only if every left ideal L of

R is of the form L = Re, where e is an idempotent of R.

Therefore, one can use the idempotents to characterize the decomposition

of semisimple rings as direct sums of minimal left ideals. We refer to [PMS02,

Theorem 2.5.11] for a proof.

Theorem 1.7.2

Let R =⊕t

i=1 Li be a decomposition of a semisimple ring as a direct sum

of minimal left ideals. There exists a family {e1, . . . , et} of elements of R

such that: each ei is a non-zero idempotent element, if i 6= j, then eiej = 0,

e1 + · · · + et = 1 and each ei can not be written as ei = e′i + e′′i , where

e′i, e′′i are idempotents such that e′i, e

′′i 6= 0 and e′ie

′′i = 0, for all 1 ≤ i ≤ t.

Conversely, if there exists a family of idempotents {e1, . . . , et} satisfying

the previous conditions, then the left ideals Li = Rei are minimal and

R =⊕t

i=1 Li.

Such a set of idempotents is called a complete set of orthogonal primitive

idempotents of the ring R. Note that such a set is not uniquely determined.

14


Example 1.7.3 Consider the matrix ring M2(F ) over a field F . The setsßÅ1 0

0 0

ã,

Å0 0

0 1

ã™and ßÅ

1 1

0 0

ã,

Å0 −1

0 1

ã™are two different complete sets of orthogonal primitive idempotents.

Given a decomposition of a semisimple ring R as a direct sum of minimal left

ideals, one can group isomorphic left ideals together. The sum of all left ideals

isomorphic to one in the decomposition, turns out to be a minimal two-sided

ideal of R, which is simple as a ring. Also, the decomposition of R as direct

sums of two-sided ideals is related to a family of idempotents. (See [PMS02,

Theorem 2.6.8].)

Theorem 1.7.4

Let R =⊕s

i=1Ai be a decomposition of a semisimple ring as a direct

sum of minimal two-sided ideals. There exists a family {e1, . . . , es} of

elements of R such that: each ei is a non-zero central idempotent element,

if i 6= j, then eiej = 0, e1 + · · · + es = 1 and each ei can not be written

as ei = e′i + e′′i , where e′i, e′′i are central idempotents such that e′i, e

′′i 6= 0

and e′ie′′i = 0, 1 ≤ i ≤ s.

The elements {e1, . . . , es} are called the primitive central idempotents of R and

they give rise to the well known Wedderburn-Artin Theorem.

Theorem 1.7.5 (Wedderburn-Artin)

A ring R is semisimple if and only if it is a direct sum of matrix rings over

division rings.

From now on, G denotes an arbitrary finite group and F a field such that

FG is semisimple. By Maschke’s Theorem (Theorem 1.6.2) this is equivalent

with saying that the order of G is coprime to the characteristic of F .

The following example is given by Charles W. Curtis and Irving Reiner in

[CR81, Example 7.39].

Example 1.7.6 Let D2n denote the dihedral group of order 2n, i.e.

D2n =⟨a, b : an = 1, b2 = 1, bab = a−1

⟩.

15

preliminaries

We have

QD2n '⊕d|n

Ad,

where

Ad =

ßQ⊕Q d = 1, 2

M2(Q(ζd + ζ−1d )) d 6= 1, 2.

The classical method for computing primitive central idempotents in FG

involves characters of the group G. All the characters of any finite group are

assumed to be characters in F , a fixed algebraic closure of the field F . For

an irreducible character χ of G, e(χ) = χ(1)|G|

∑g∈G χ(g−1)g is the primitive

central idempotent of FG associated to χ and eF (χ) is the only primitive

central idempotent e of FG such that χ(e) 6= 0. The field of character values

of χ over F is defined as F (χ) = F (χ(g) : g ∈ G), that is, the field extension

of F generated over F by the image of χ. The automorphism group Aut(F )

acts on FG by acting on the coefficients, that is, σ∑g∈G agg =

∑g∈G σ(ag)g,

for σ ∈ Aut(F ) and ag ∈ F . Following Toshihiko Yamada [Yam74], one knows

that

eF (χ) =∑

σ∈Gal(F (χ)/F )

σe(χ). (2)

The known methods to compute the character table of a finite group are very

time consuming. Therefore, in practical applications, the classical description

of primitive central idempotents sometimes is of limited use. New methods

for the computation of the primitive central idempotents in a group algebra

do not involve characters. The main ingredient in this theory is the following

element, introduced in [JLP03] by Eric Jespers, Guilherme Leal and Antonio

Paques. If K �H ≤ G, then let ε(H,K) be the element of QH ⊆ QG defined

as

ε(H,K) =

® “K if H = K,∏M/K∈M(H/K)(

“K − M) if H 6= K,

where M(H/K) denotes the set of minimal normal non-trivial subgroups of

H/K. Furthermore, e(G,H,K) denotes the sum of the different G-conjugates

of ε(H,K).

Aurora Olivieri, Angel del Rıo and Juan Jacobo Simon achieved a break-

through in the study of primitive central idempotents associated to monomial

characters in [OdRS04]. We will explain this in detail.

16


Let H be a subgroup of G and χ a character of H. Then χG is given by

χG(g) =1

|H|∑x∈G

χ◦(xgx−1),

where χ◦ is defined by χ◦(h) = χ(h), if h ∈ H, and χ◦(y) = 0 if y /∈ H. It is

well known that χG is a character of G and it is called the induced character

on G (see [Isa76, Corollary 5.3]). A (complex) character ψ of G is called

monomial if there exists a subgroup H ≤ G and a linear character χ of H such

that ψ = χG. The group G is called monomial if all its irreducible (complex)

characters are monomial.

Definition 1.7.7

A pair (H,K) of subgroups of G is called a Shoda pair if it satisfies the

following conditions:

1. K �H;

2. H/K is cyclic;

3. if g ∈ G and [H, g] ∩H ⊆ K, then g ∈ H.

Now, one can rephrase an old theorem of Kenjiro Shoda [Sho33] as follows.

Proposition 1.7.8 (Shoda)

If χ is a linear character of a subgroup H of G with kernel K, then the

induced character χG is irreducible if and only if (H,K) is a Shoda pair.

For monomial groups G, Shoda pairs are sufficient to be able to construct

all primitive central idempotents of QG, see [OdRS04, Corollary 2.3].

Theorem 1.7.9 (Olivieri-del Rıo-Simon)

A finite group G is monomial if and only if every primitive central idem-

potent of QG is of the form αe(G,H,K), for α ∈ Q and (H,K) a Shoda

pair of G.

So all primitive central idempotents of QG for monomial groups correspond

to Shoda pairs of G. However, different Shoda pairs can contribute to the same

primitive central idempotent of G. This is expressed in [OdRS06, Proposition

1.4] in terms of a relation on Shoda pairs.

17

preliminaries

Proposition 1.7.10 (Olivieri-del Rıo-Simon)

Let (H1,K1) and (H2,K2) be two Shoda pairs of a finite group G and

consider α1, α2 ∈ Q such that ei = αie(G,Hi,Ki) is a primitive central

idempotent of QG, for i = 1, 2. Then e1 = e2 if and only if a g ∈ G exists

such that Hg1 ∩K2 = Kg

1 ∩H2.

One says that a set S of Shoda pairs of G is a complete and non-redundant

set of Shoda pairs of G if all primitive central idempotents of G can be realized

by a pair in S and no two different pairs in S determine the same idempotent

in QG.

Note that α is uniquely determined by the fact that αe(G,H,K) is an idem-

potent. If for example, the G-conjugates of ε(H,K) are orthogonal, then α = 1.

This happens in the case when G is a strongly monomial group. For this one

introduces strong Shoda pairs.

Definition 1.7.11

A strong Shoda pair of G is a pair (H,K) of subgroups of G satisfying

the following conditions:

1. K ≤ H �NG(K);

2. H/K is cyclic and a maximal abelian subgroup of NG(K)/K;

3. for every g ∈ G \NG(K), ε(H,K)ε(H,K)g = 0.

A (complex) character ψ of G is said to be strongly monomial if there is

a strong Shoda pair (H,K) of G and a linear character χ of H with kernel

K such that ψ = χG. A group G is strongly monomial if all its irreducible

(complex) characters are strongly monomial.

The big advantage of working with strongly monomial groups is that one

can describe the structure of the simple components of QG. The following

results can be found in [OdRS04, Propositions 3.3 and 3.4, Theorem 4.7].


The following are equivalent for a pair (H,K) of subgroups of G:

1. (H,K) is a strong Shoda pair of G;

2. (H,K) is a Shoda pair of G, H � NG(K) and the G-conjugates of

ε(H,K) are orthogonal.

18


Moreover, if the previous conditions hold then CenG(ε(H,K)) = NG(K)

and e(G,H,K) is a primitive central idempotent of QG.


Let (H,K) be a strong Shoda pair of a finite group G and let k = [H : K],

N = NG(K), n = [G : N ], yK a generator of H/K and ψ : N/H →N/K a left inverse of the canonical projection N/K → N/H. The simple

algebra QGe(G,H,K) is isomorphic to Mn(Q(ζk)∗ατ N/H) and the action

and twisting are given by

αnH(ζk) = ζik, if yKψ(nH) = yiK and

τ(nH, n′H) = ζjk, if ψ(nn′H)−1ψ(nH)ψ(n′H) = yjK,

for nH, n′H ∈ N/H and integers i and j.


A finite group G is strongly monomial if and only if every primitive central

idempotent of QG is of the form e(G,H,K), for (H,K) a strong Shoda

pair of G.

In analogue to the Shoda pairs, one says that a set S of strong Shoda pairs

of G is a complete and non-redundant set of strong Shoda pairs of G if all

primitive central idempotents of G can be realized by a pair in S and no two

different pairs in S determine the same idempotent in QG.

The following is proven in [OdR03, Proposition 2.4].

Proposition 1.7.15 (Olivieri-del Rıo)

If (H,K) is a strong Shoda pair of a finite group G and N = NG(K), then

N/H is isomorphic to a subgroup of the group of units of Z/[H : K]Z and

in particular N/H is abelian.

Let k be a positive integer and Ck = 〈c〉 a cyclic group of order k. Then

there are isomorphisms

Gal(Q(ζk)/Q) → U(Z/kZ) → Aut(Ck)

(ζk 7→ ζrk) 7→ r 7→ (σr : c→ cr)

Throughout this thesis we will abuse the notation and consider these isomor-

phisms as equalities. For example, a subgroup H of Aut(Ck) will be identified

19

preliminaries

with a subgroup of U(Z/kZ) and with Gal(Q(ζk)/Q(ζk)H). In particular, for a

strong Shoda pair (H,K) of G, with the notation of Proposition 1.7.13 and be-

cause H/K is a maximal abelian subgroup of N/K, the action α of the crossed

product Q(ζk)∗ατ N/H is faithful. Therefore, the crossed product Q(ζk)∗ατ N/Hcan be described as a classical crossed product (Q(ζk)/F, τ), where F is the

center of the algebra which is determined by the action of N/H on H/K. In

this way, the Galois group Gal(Q(ζk)/F ) can be identified with N/H and with

this identification F = Q(ζk)N/H .

Examples of strongly monomial groups are abelian-by-supersolvable groups

[OdRS04, Corollary 4.6]. All monomial groups of order smaller than 1000 are

strongly monomial and the smallest monomial non-strongly monomial group

is a group of order 1000, number 86 in the library of the GAP system [Olt07,

GAP14].

Metabelian groups are also strongly monomial and provide an easier descrip-

tion for the strong Shoda pairs.

Theorem 1.7.16 (Olivieri-del Rıo-Simon)

Let G be a finite metabelian group and let A be a maximal abelian sub-

group of G containing the commutator subgroup G′. The primitive central

idempotents of QG are the elements of the form e(G,H,K), where (H,K)

is a pair of subgroups of G satisfying the following conditions:

1. H is a maximal element in the set

{B ≤ G : A ≤ B and B′ ≤ K ≤ B};

2. H/K is cyclic.

Theorem 1.7.16 and Proposition 1.7.13 allow one to easily compute the primi-

tive central idempotents and the Wedderburn components of the rational group

algebra of a finite metacyclic group. Recall that this is a group G having a

normal cyclic subgroup N = 〈a〉 such that G/N = 〈bN〉 is cyclic. Every finite

metacyclic group G has a presentation of the form

G =⟨a, b : am = 1, bn = at, ab = ar

⟩,

where m,n, t, r are integers satisfying the conditions rn ≡ 1 mod m and m

divides t(r − 1). Define σ ∈ Aut(〈a〉) as σ(a) = ab = ar. When u is the order

of σ, it follows that u | n. For every d | u, let Gd = 〈a, bd〉.

20


Lemma 1.7.17

With notations as above, the primitive central idempotents of QG are the

elements of the form e(G,Gd,K) where d is a divisor of u and K is a

subgroup of Gd satisfying the following conditions:

1. d = min{x | u : arx−1 ∈ K};

2. Gd/K is cyclic.

We apply this result to the specific case of split metacyclic groups with

trivial kernel.

Corollary 1.7.18

Let p and q be different prime numbers, m and n positive integers and

G = 〈a〉o1〈b〉, with |a| = qm and |b| = pn. A complete and non-redundant

set of strong Shoda pairs of G consists of two types:

1.ÄG,Li :=

ä, bp

i∂ä, i = 0, . . . , n;

2.Ä〈a〉 ,Kj :=

äqj∂ä, j = 1, . . . ,m.

The simple components of QG are:

(i) QGε (G,Li) ' Q(ζpi), i = 0, . . . , n;

(ii) QGε (〈a〉 ,Kj) ' Q(ζqj)∗ 〈b〉 'Mpn

(Q(ζqj )

〈b〉) , j = 1, . . . ,m.

Proof From the notation 〈a〉o1 〈b〉, it automatically follows that Cen〈b〉(a) =

1, or equivalently, that 〈b〉 acts faithfully on 〈a〉. Let σ be the automorphism

of 〈a〉 given by σ(a) = ab and assume that σ(a) = ar, with r ∈ Z. As the

kernel of the restriction map Aut(〈a〉) → AutÄäqm−1∂ä

has order qm−1, it

intersects 〈σ〉 trivially and therefore the restriction of σ toäqm−1∂

also has

order pn. This implies that q ≡ 1 mod pn and thus q is odd. Therefore,

AutÄäqj∂ä

(= U(Z/qjZ)) is cyclic for every j = 0, 1, . . . ,m and 〈σ〉 is the

unique subgroup of Aut(〈a〉) of order pn. Hence, for every i = 1, . . . ,m, the

image of r in Z/qiZ generates the unique subgroup of U(Z/qiZ) of order pn.

In particular rpn ≡ 1 mod qm and rp

j 6≡ 1 mod q for every j = 0, . . . , n − 1.

21

preliminaries

Therefore, r 6≡ 1 mod q and hence G′ =⟨ar−1

⟩= 〈a〉. Using the description

of strong Shoda pairs of G as given in Lemma 1.7.17 and the description of

the associated simple algebra given in Proposition 1.7.13 and Theorem 1.5.2,

we obtain the result.

Although the description of the Wedderburn decomposition and the primi-

tive central idempotents of QG is known for strongly monomial groups G, it is

not trivial to obtain a description of a complete set of (non-central) orthogo-

nal primitive idempotents. However, Eric Jespers, Gabriela Olteanu and Angel

del Rıo obtained such a description in [JOdR12, Theorem 4.5] for nilpotent

groups.

Let G be a finite nilpotent group, one writes G =∏p Sp, i.e. the direct

product of its Sylow p-subgroups Sp for p dividing |G|. The 2-part of G is the

Sylow 2-subgroup S2. The 2′-part is the direct product∏p 6=2 Sp of all Sylow

p-subgroups for p 6= 2.

Theorem 1.7.19 (Jespers-Olteanu-del Rıo)

Let G be a finite nilpotent group and (H,K) a strong Shoda pair of G.

Set e = e(G,H,K), ε = ε(H,K), H/K = 〈a〉, N = NG(K) and let N2/K

and H2/K = 〈a2〉 (respectively N2′/K and H2′/K = 〈a2′〉) denote the

2-parts (respectively 2′-parts) of N/K and H/K respectively. The group

〈a2′〉 has a cyclic complement 〈b2′〉 in N2′/K.

A complete set of orthogonal primitive idempotents of QGe consists

of the conjugates of b2′β2ε by the elements of T2′T2TG/N , where T2′ =

{1, a2′ , a22′ , . . . , a

[N2′ :H2′ ]−12′ }, TG/N denotes a right transversal of N in G

and β2 and T2 are given according to the cases below.

1. If H2/K has a complement M2/K in N2/K, then β2 = ”M2. More-

over, if M2/K is cyclic, there exists b2 ∈ N2 such that N2/K is given

by the following presentation

〈a2, b2 : a22n = b2

2k = 1, a2b2 = a2

r〉,

and if M2/K is not cyclic, there exist b2, c2 ∈ N2 such that N2/K

is given by the following presentation

〈a2, b2, c2 : a22n = b2

2k = 1, c22 = 1, a2

b2 = a2r,

a2c2 = a2

−1, [b2, c2] = 1〉,

with r ≡ 1 mod 4 (or equivalently a22n−2

is central in N2/K).

22


a) T2 = {1, a2, a22, . . . , a

2k−12 }, if a2

2n−2

is central in N2/K (unless

n ≤ 1) and M2/K is cyclic; and

b) otherwise

T2 = {1, a2, a22, . . . , a

[N2:H2]/2−12 , a2n−2

2 , a2n−2+12 , . . . ,

a2n−2+[N2:H2]/2−12 }.

2. If H2/K does not have a complement in N2/K, there exist elements

b2, c2 ∈ N2, such that N2/K is given by the following presentation

〈a2, b2, c2 : a22n = b2

2k = 1, c22 = a2

2n−1

, a2b2 = a2

r,

a2c2 = a2

−1, [b2, c2] = 1〉,

with r ≡ 1 mod 4 and we set m = [H2′ : K]/[N2′ : H2′ ].

a) β2 = “b2 and T2 = {1, a2, a22, . . . , a

2k−12 }, if either m = 1 or the

order of 2 modulo m is odd and n− k ≤ 2 and

b) β2 = “b2 1+xa2n−2

2 +ya2n−2

2 c22 and

T2 = {1, a2, a22, . . . , a

2k−12 , c2, c2a2, c2a

22, . . . , c2a

2k−12 }

with

x, y ∈ QÄa

[N2′ :H2′ ]2′ , a2k

2 + a−2k

2

ä,

satisfying (1 + x2 + y2)ε = 0, if m 6= 1 and either the order of

2 modulo m is even or n− k > 2.

For rational group algebras of arbitrary finite metacyclic groups, we would

like to be able to give a similar description of a complete set of orthogonal

primitive idempotents as in the previous theorem. Unfortunately, the approach

of Theorem 1.7.19 does not apply here. To show this, the following example

was given in [JOdR12, Remark 4.8].

Example 1.7.20 Let G = C7 o1 C3 = 〈a〉 o1 〈b〉, with b−1ab = a2 and e =

e(G, 〈a〉, 1) = ε(〈a〉, 1). One can check that there does not exist a complete set

of orthogonal primitive idempotents of QGe = Q(ζ7)∗〈b〉 = (Q(ζ7)/Q(√−7), 1)

formed by Q(a)-conjugates of bε.

23

preliminaries

Nevertheless, in Chapter 2, and more particular in Corollary 2.2.5, we show

how to overcome the difficulties and we will produce a complete set of orthog-

onal primitive idempotents for Q(C7 o1 C3).

1.8 Z-orders

We collect some basic results about Z-orders from the book of Sudarshan

Sehgal [Seh93].

A subring O of a finite dimensional Q-algebra A is called a Z-order (or

order) if it is a finitely generated Z-module such that QO = A. For example,

the ring of integers R of a number field F is an order in F . If G is a finite

group, then ZG is an order in QG and Z (ZG) is an order in Z (QG). More

generally, RG is an order in F G. If O is an order in a division ring D,

then Mn (O) is an order in Mn (D). An order is said to be maximal if it is

not properly contained in a bigger order, within the same finite dimensional

Q-algebra.

The following lemmas provide examples of maximal orders and can be found

in [Rei75, Theorem 8.6] and [Seh93, Lemmas 4.4 and 4.5].

Lemma 1.8.1

The ring of integers of a number field F is the unique maximal order in

F .

Lemma 1.8.2

Let A be a finite dimensional Q-algebra with Wedderburn decomposition⊕iAei. Let O be a maximal order in A, then

1. O =⊕

iOei. Moreover, Oei is a maximal order of Aei;

2. Mn(O) is a maximal order in Mn(A).

Lemma 1.8.3

Every order in A is contained in a maximal order.

Example 1.8.4 Let G be a finite abelian group and QG =⊕

dQ(ζd) with

possible repetition. Then⊕

d Z[ζd] is the unique maximal order in QG and

hence ZG ⊆⊕

d Z[ζd].

24

1.8 Z-orders

Let R be a ring. The unit group of R is denoted U(R). The following lemmas

can be found in [Seh93, Lemmas 4.2 and 4.6]

Lemma 1.8.5

The intersection of two orders in A is again an order in A.

Lemma 1.8.6

Suppose O1 ⊆ O2 are two orders in A. The index of their unit groups

[U(O2) : U(O1)] is finite. Furthermore, if u ∈ O1 is invertible in O2, then

u−1 ∈ O1.

Therefore, for two arbitrary orders O1,O2 in A, we have that the index

[U(O2) : U(O1 ∩ O2)] is finite, in other words U(O1) and U(O2) are commen-

surable.

The following theorem on orders follows from deep results by Carl Ludwig

Siegel in [Sie43] and Armand Borel and Harish-Chandra in [BHC62].

Theorem 1.8.7 (Siegel-Borel-Harish-Chandra)

The group of units of an order of a finite dimensional Q-algebra is finitely

generated.

It follows that the unit groups U(ZG) and Z(U(ZG)) = U(Z(ZG)) are

finitely generated if G is a finite group.

Let Γ be a finitely generated abelian group. The rank of all free abelian

subgroups of finite index in Γ is an invariant of the group, called the rank of Γ.

Note that replacing generators of Γ by powers of themselves yields generators

of a subgroup of finite index. We will use this fact throughout this document

without explicitly mentioning it.

As an immediate application of Theorem 1.8.7, one obtains that U(R) is

finitely generated, when R is the ring of integers in a number field. One can

even determine the rank of U(R). This is one of the most famous results on

units of orders.

Theorem 1.8.8 (Dirichlet’s Unit Theorem)

Let F be a number field of degree n = r + 2s over Q, where r is the

number of real embeddings of F and s is the number of pairs of complex

embeddings of F . Let R be the ring of integers of F . The unit group U(R)

is a finitely generated abelian group. Moreover, U(R) = C × A, where C

is a finite cyclic group and A is torsion free of rank r + s− 1.

25

preliminaries

1.9 congruence subgroup problem

Let K be a number field and A a finite dimensional semisimple K-algebra.

Let F be a splitting field of A over K and fix an isomorphism h : F ⊗K A→Mn1

(F ) × · · · ×Mnm(F ) of F -algebras. Let hi : F ⊗K A → Mni(F ) be the

composition of h with the projection on the i-th component. By definition,

the reduced norm of a ∈ A is given by nrdA/K(a) =∏mi=1 det(hi(1⊗ a)).

Examples 1.9.1 Let K be a number field.

1. If Gal(K/Q) = {σ1, . . . , σn}, then one has the following isomorphism

K ⊗Q K → Kn : x⊗ y 7→ (xσi(y))1≤i≤n.

Therefore nrdK/Q(x) =∏ni=1 σi(x).

2. If A is the quaternion algebraÄa,bK

äwith splitting field E, then the

following is an isomorphism of E-algebras:

E ⊗K A → M2(E)

1⊗ (x+ y√a+ z

√b+ t

√ab) 7→

Çx+ y

√a z

√b+ t

√ab

z√b− t

√ab x− y

√a

åand nrdA/K(x+ y

√a+ z

√b+ t

√ab) = x2 − ay2 − bz2 + abt2.

In particular, let K be a number field and D a division algebra which is

finite dimensional over K with degree m. If A = Mn(D) and O is an order

in D, then, one can prove that nrdA/K(A) ⊆ K and nrdA/K(Mn(O)) ⊆ Z(O)

(see [CR81, Corollary 26.2]). Also, a ∈ A is a unit in A if and only if nrdA/K(a)

is invertible (see [CR81, page 170]). One denotes by GLn(O) the invertible

matrices in Mn(O), and by SLn(O) the matrices of reduced norm 1 in GLn(O).

It is easy to verify that nrdA/K(Z(GLn(O))) = U(Z(O))nm. By Dirichlet’s

Unit Theorem 1.8.8, U(Z(O)) is a finitely generated abelian group, hence

[GLn(O) : Z(GLn(O))SLn(O)] ≤ [U(Z(O)) : U(Z(O))nm] is finite.

Let m be a positive integer, one calls

SLn(O,m) = {M ∈ SLn(O) : M − In ∈Mn(mO)}

the congruence subgroup of level m. Clearly, SLn(O,m) has finite index in

SLn(O). The Congruence Subgroup Problem (CSP) asks for the converse of

this statement:

26


Does every subgroup of SLn(O) of finite index

contain a congruence subgroup?

For a more detailed survey on the Congruence Subgroup Problem, see for

example [Sur03, PR10].

More relevant for our investigations, is the following problem. Let I be a

non-zero ideal of O. Matrices having 1 at every entry on the diagonal and only

one off-diagonal entry in I and 0 in all other off-diagonal entries, are called

elementary matrices modulo I. The group generated by these elementary

matrices is written as En(I). A relevant question is:

Is En(I) of finite index in SLn(O)?

This problem seems to be strongly related to the Congruence Subgroup Prob-

lem. When the CSP has an “almost” positive answer, it is proven that indeed

En(I) is of finite index in SLn(O). The explanation of what an “almost” pos-

itive answer to CSP is, is out of the scope of this thesis. Again, the interested

reader is referred to [Sur03, PR10].

Leonid N. Vasersteın and, independently Jacques Tits, have proven a posi-

tive answer to the question for n ≥ 3 [Vas73, Tit76] and Tyakal Nanjundiah

Venkataramana found a proof for the case n = 2 [Ven94]. Parts of these re-

sults were proven earlier by others as well, for example by Hyman Bass [Bas64],

Leonid N. Vasersteın [Vas72] and Bernhard Liehl [Lie81]. Actually, they have

proven the statements in the setting of linear algebraic groups, but we state it

only in the context essential to our investigations.

Theorem 1.9.2 (Bass-Vasersteın-Tits)

Let O be an order in a finite dimensional rational division algebra. If

n ≥ 3, then [SLn(O) : En(I)] <∞ for every non-zero ideal I of O.

Theorem 1.9.3 (Vasersteın-Liehl-Venkataramana)

Let O be an order in a finite dimensional rational division algebra D. If

D is different from Q, a quadratic imaginary extension of Q and a totally

definite quaternion algebra with center Q, then [SL2(O) : E2(I)] <∞ for

all non-zero ideals I of O.

For the case n = 1, almost nothing is known, except for a result of Ernst

Kleinert [Kle00a].

27

preliminaries

Theorem 1.9.4 (Kleinert)

When O is an order in a non-commutative division ring D, SL1(O) is finite

if and only if D is a totally definite quaternion algebra.

The above theorems lead us to the definition of exceptional simple algebras.

Definition 1.9.5

A simple finite dimensional rational algebra is an exceptional simple alge-

bra if it is of one of the following types:


nion algebra;

EC2: a 2 × 2-matrix ring over the rationals, a quadratic imaginary ex-

tension of the rationals or over a totally definite quaternion algebra

over Q.

Let G be a finite group and F a number field. We say that FG contains

an exceptional component if one of the simple algebras in the Wedderburn

decomposition of FG is an exceptional simple algebra.

In joined work with Florian Eisele and Ann Kiefer, we showed that the

exceptional components of a group algebra FG over a number field F are of a

very restricted type, [EKVG15, Theorems 3.1 and 3.5, Proposition 3.3].

Theorem 1.9.6 (Eisele-Kiefer-Van Gelder)

If G is a finite subgroup of GL2(F ), for F a quadratic imaginary extension

of the rationals, such that G spans M2(F ) over F , then G is solvable,

|G| = 2a3b for some a, b ∈ N and F is one of the following fields:

1. Q(√−1);

2. Q(√−2);

3. Q(√−3).

Furthermore, elements of a finite subgroup G of GL2(Q) can only have

prime power orders 1, 2, 3 and 4.

The following proposition turns the classification of all finite subgroups of

GL2(F ) for F ∈{Q(√−d) : d = 0, 1, 2, 3

}into a finite problem.

28


Proposition 1.9.7 (Eisele-Kiefer-Van Gelder)

Let F ∈{Q(√−d) : d = 0, 1, 2, 3

}, and let G ≤ GL2(F ) be a finite group.

The group G can be embedded in the finite group GL(2, 25).

Theorem 1.9.8 (Eisele-Kiefer-Van Gelder)

Let G be a finite group. If G is a subgroup of GL2(D), for D a totally

definite quaternion algebra with center Q, such that G spans M2(D) over

Q, then D is one of the following algebras:

1.Ä−1,−1

Q

ä;

2.Ä−1,−3

Q

ä;

3.Ä−2,−5

Q

ä.

Due to Theorems 1.9.6 and 1.9.8, the exceptional simple algebras appearing

in a group algebra restrict to the following.

Corollary 1.9.9

If a simple finite dimensional rational algebra is an exceptional component

of some group algebra FG for some number field F , then it is of one of

the following types:


nion algebra;

EC2: M2(Q), M2(Q(√−1)), M2(Q(

√−2)), M2(Q(

√−3)), M2

Ä−1,−1

Q

ä,

M2

Ä−1,−3

Q

ä, M2

Ä−2,−5

Q

ä.

Note that, Theorems 1.9.6 and 1.9.8 could also have been reduced from clas-

sifications of Behnam Banieqbal [Ban88] and Gabriele Nebe [Neb98, Theorems

6.1 and 12.1], however their proofs are very long and tedious. Therefore, we

provided elementary proofs in [EKVG15] for the very specific cases were we

want to use Banieqbal’s or Nebe’s results.

Let A be a finite dimensional rational simple algebra and denote the absolute

value of the reduced norm nrdA/Q : A → Q by NA/Q : A → Q≥0. For any

order O of A, we have NA/Q(O) ⊆ N ∪ {0}.

29

preliminaries

Examples 1.9.10

1. Let d be a square-free positive integers, then NQ(√−d)/Q(x + y

√−d) =

x2 + dy2 for each x+ y√−d ∈ Q(

√−d).

2. Let a, b be negative rational numbers and let A be the quaternion algebraÄa,bQ

ä, then NA/Q(x + y

√a + z

√b + t

√ab) = x2 − ay2 − bz2 + abt2 for

each x+ y√a+ z

√b+ t

√ab ∈ A.

One says that an order O in A is left norm Euclidean if for every x, y ∈ O,

with y 6= 0, there exist q, r ∈ O such that x = qy + r with N(r) < N(y). It is

easy to show that a left norm Euclidean order necessarily is a maximal order,

for a proof see for example [CCL13, Proposition 2.8].

Theorems 1.9.6 and 1.9.8 show that if M2(D) is an exceptional component

of a group algebra FG, then D is either a field Q(√−d), with d ∈ {0, 1, 2, 3},

or a quaternion algebraÄa,bQ

ä, with (a, b) ∈ {(−1,−1), (−1,−3), (−2,−5)}. It

is well known that in the 4 commutative cases, the ring of integers O of D is

a Euclidean domain (see for example [Wei98, Proposition 6.4.1]), and actually

the reduced norm turns O into a (left) norm Euclidean order. Moreover, the

listed quaternion algebras are the only possible totally definite quaternion

algebras over Q having a left norm Euclidean order O and O is the unique

maximal order (see [Fit12, Theorem 2.1] or [CCL13, Theorem 1.6]).

Corollary 1.9.11 (Eisele-Kiefer-Van Gelder)

Let G be a finite group and F a number field with the property that FG

contains an exceptional component M2(D). Then D contains a left norm

Euclidean order O, which is necessarily the unique maximal order of D.

The corresponding maximal orders are listed in Table 1 below.

Table 1: Maximal orders

Division ring Maximal order

Q ZQ(√−1) Z[

√−1]

Q(√−2) Z[

√−2]

Q(√−3) Z

î1+√−3

2

ócontinued

30

1.10 finite subgroups of exceptional simple algebras

Division ring Maximal orderÄ−1,−1

Q

äZ[1, i, j, 1

2 (1 + i+ j + ij)]Ä−1,−3

Q

äZ[1, i, 1

2 (1 + j), 12 (i+ ij)]Ä

−2,−5Q

äZ[1, 1

4 (2 + i− ij), 14 (2 + 3i+ ij), 1

2 (1 + i+ j)]

1.10 finite subgroups of exceptional simple algebras

We recall the classification of finite subgroups of division rings from Shimshon

Avraham Amitsur [Ami55]. This classification splits into 2 parts: a list of

some Z-groups and a list of non-Z-groups. Recall that Z-groups are groups

for which all Sylow p-subgroups are cyclic. We use notations from [SW86,

Theorems 2.1.4, 2.1.5] and [CdR14, Theorem 2.2].

Theorem 1.10.1 (Amitsur)

(Z) The Z-groups which are finite subgroups of division rings are:

a) the finite cyclic groups;

b) Cm o2 C4, with m odd and C4 acting by inversion on Cm;

c) Cm ok Cn, with n 6= 1, gcd(m,n) = 1 and, using the following

notation

Pp = Sylow p-subgroup of Cm;

Qp = Sylow p-subgroup of Cn;

Xp = {q | n : q prime and [Pp, Qq] 6= 1};

Rp =∏q∈Xp

Qq;

we assume that Cn =∏p|mRp and the following properties

hold for every prime number p | m and q ∈ Xp:

i. q · oqvq(k)(p) - o mn|Pp| |Rp|

(p);

ii. if q is odd or p ≡ 1 mod 4, then vq(p− 1) ≤ vq(k);

iii. if q = 2 and p ≡ −1 mod 4, then v2(k) is either 1 or

greater than v2(p+ 1).

31

preliminaries

(NZ) The finite subgroups of division rings which are not Z-groups are:

a) O∗ =⟨s, t : (st)2 = s3 = t4

⟩(binary octahedral group);

b) SL(2, 5);

c) Q4k with k even;

d) SL(2, 3)×M , with M a group in (Z) of order coprime to 6 and

o|M |(2) odd;

e) Q8×M , with M a group in (Z) of odd order such that o|M |(2)

is odd.

Remark 1.10.2 Assume that G = Cm ok Cn is a group satisfying the hypo-

thesis in (Z)(c) in Theorem 1.10.1. Since G is not cyclic, Cn acts non-trivially.

Let p be a prime divisor of m, such that Cn acts non-trivially on the Sylow

p-subgroup Pp of Cm. Let q1, . . . , qh be the prime divisors of n such that for

each qi, Qqi acts non-trivially on Pp, so Rp = Qq1 · · ·Qqh . Let kp be the order

of the kernel of the action of Rp on Pp. Then Pp okp Rp is a direct factor of

G. From the conditions on (Z)(c) it follows that for every prime divisor qi of

|Rp|, we have vqi(p− 1) ≤ vqi(k). Since Cn =∏p|mRp is a direct product, all

Xp are mutually disjoint, k =∏p|m kp and kp is only divisible by q if q ∈ Xp.

Therefore vqi(k) = vqi(kp).

Let pγ be the order of Pp, and let the action of Rp on Pp be defined by

σ : Rp → Aut(Pp), then Rp/Ker(σ) ' Im(σ). It follows that|Rp|kp

divides

|Aut(Pp)| = pγ−1(p− 1). Thus,|Rp|kp

divides p− 1.

On the other hand, for all 1 ≤ i ≤ h, let qβii and qαii be the order of Qqiand the order of the kernel of the action of Qqi on Pp, respectively. Hence,

|Rp| = qβ1

1 · · · qβhh and kp = qα1

1 · · · qαhh . Given a fixed i, as Pp oqαi

iQqi is not

cyclic, βi 6= αi, so that qi divides|Rp|kp

. It now follows that

1 ≤ vqiÅ |Rp|kp

ã≤ vqi(p− 1) ≤ vqi(kp). (3)

Note that ones deduce from (3) that every prime divisor of |Rp| is a prime

divisor of both|Rp|kp

and kp. In particular, this means that any prime divisor

of n divides both k and nk . Finally, by (3) every αi is not zero, this implies

that no Qqi acts faithfully on Pp, and neither does Rp. Hence kp 6= 1, and in

particular k 6= 1.

32

1.11 cyclotomic units

Gabriele Nebe gave a list of maximal finite subgroups of GLn(D), spanning

Mn(D) over Q, for D a totally definite quaternion algebra with center of degree

d over Q, such that nd ≤ 10. We recall this result for the case n = 2 and d = 1,

see [Neb98, Theorems 6.1 and 12.1]. This result could also have been distilled

from the classification of Banieqbal [Ban88], who classified all finite subgroups

of 2× 2-matrices over division rings of characteristic zero.

We list the groups in the next theorem using their GAP identification number.

Theorem 1.10.3 (Nebe)

The maximal finite subgroups of 2×2-matrices over totally definite quater-

nion algebras D with center Q, spanning M2(D) over Q, are:

[144, 124], [144, 128], [240, 89], [240, 90], [288, 389],

[720, 409], [1152, 155468] and [1920, 241003].

Alternatively, Eric Jespers and Angel del Rıo were able to obtain this list of

finite subgroups of 2×2-matrices over totally definite quaternion algebras with

center Q in [JdR], by describing the Sylow subgroups of such groups. This can

be done only by using strong Shoda pairs and some elementary techniques.

We conclude by studying finite subgroups of U(B) for exceptional compo-

nents B of group algebras. Let G be a finite subgroup of U(B). When B

is a division ring, G appears in Amitsur’s classification, see Theorem 1.10.1.

When B is an exceptional 2× 2-matrix ring M2(Q(√−d)) with d ∈ {0, 1, 2, 3},

G is a subgroup of GL(2, 25) by Proposition 1.9.7. If B is one of M2

Ä−1,−1

Q

ä,

M2

Ä−1,−3

Q

äor M2

Ä−2,−5

Q

ä, then G appears as a subgroup of the groups in

the classification of Gabriele Nebe, see Theorem 1.10.3.

1.11 cyclotomic units

We are interested in U(ZG), and because of the Wedderburn decomposition,

also in U(Z[ζn]). In this section, we deal with units in Z[ζn].

If n > 1 and k is an integer coprime to n, then one defines

ηk(ζn) =1− ζkn1− ζn

= 1 + ζn + ζ2n + · · ·+ ζk−1

n ,

and one can check that ηk(ζn) is a unit of Z[ζn]. One extends this notation by

defining

ηk(1) = 1.

33

preliminaries

The units of the form ηk(ζjn), with j, k and n integers, such that gcd(k, n) = 1,

are called the cyclotomic units of Q(ζn).

The next lemma is easy to verify.

Lemma 1.11.1

Let n > 1 and both k and k1 coprime with n. The cyclotomic units satisfy

the following equalities:

(1) ηk(ζn) = ηk1(ζn), if k ≡ k1 mod n;

(2) ηkk1(ζn) = ηk(ζn)ηk1(ζkn);

(3) η1(ζn) = 1;

(4) ηn−k(ζn) = −ζ−kn ηk(ζn).

A classical result, which goes back to the work of Ernst Eduard Kummer

from the 19th century, states that the cyclotomic units of Q(ζn) generate a

subgroup of finite index in U(Z[ζn]), see for example [Was82, p. 147]. Further-

more, when n is a power of a prime number, one knows the following basis

(see [Was82, Theorem 8.2]).

Theorem 1.11.2 (Kummer)

Let p be a prime number and n a non-negative integer. The setßηk(ζpn) : 1 < k <

pn

2, p - k

™generates a free abelian subgroup of finite index in U(Z[ζpn ]).

The cyclotomic units of QG are, by definition, all the elements of QG which

project to a cyclotomic unit in a commutative Wedderburn component Q(ζn)

for some n ∈ N and project to 1 in the remaining components.

1.12 bass units

Consider a finite group G, an element g ∈ G of order n, and suppose that k

and m are positive integers for which km ≡ 1 mod n. One verifies that

uk,m(g) = (1 + g + · · ·+ gk−1)m +1− km

n(1 + g + g2 + · · ·+ gn−1)

34

1.12 bass units

is a unit of the integral group ring ZG. The units of this form were introduced

by Hyman Bass in [Bas66] and are known as Bass units or Bass cyclic units.

Note that Bass units of Z 〈g〉 project to powers of cyclotomic units in the

Wedderburn components of Q 〈g〉. We denote by B1(G) the group generated

by the Bass units of ZG. Bass proved that, if G is a cyclic group, then B1(G)

is a subgroup of finite index in U(ZG). Hyman Bass and John Willard Milnor

extended this result to finite abelian groups using K-theory to reduce to the

cyclic case. If R is a ring, then GL(R) is defined as⋃n≥1 GLn(R) and E(R) as⋃

n≥1En(R). By K1(R) one denotes GL(R)/E(R). A deep result of Hyman

Bass proves that the rank of K1(ZG) equals the rank of Z(U(ZG)) and that

the image of B1(G) in K1(ZG) is of finite index in K1(ZG) [Bas66, Corollary

6.3, Theorem 5]. Applied to finite abelian groups, this gives the following.

Theorem 1.12.1 (Bass-Milnor)

When G is a finite abelian group, the group B1(G) is a subgroup of finite

index in U(ZG).

We will refer to this result as the Bass-Milnor Theorem while the Bass Theorem

refers to the result for cyclic groups. The Bass Theorem also provides a basis

consisting of Bass units for a free abelian subgroup of finite index in U(ZG),

provided that G is cyclic. As far as we know, no basis consisting of Bass units

for a free abelian subgroup of finite index was known for an arbitrary abelian

group G. In Chapter 4, we give a constructive K-theoretic-free proof of the

Bass-Milnor Theorem and describe a basis formed by Bass units for a free

abelian subgroup of finite index in U(ZG) for finite abelian groups G.

Jairo Goncalves and Donald Passman showed the following equalities in

[GP06, Lemma 3.1], inspired by those for cyclotomic units from Lemma 1.11.1.

Lemma 1.12.2 (Goncalves-Passman)

Let g ∈ G, n = |g| and k, k1,m,m1 be positive integers such that km ≡km1 ≡ km1 ≡ 1 mod n. The Bass units satisfy the following equalities:

(1) uk,m(g) = uk1,m(g), if k ≡ k1 mod n;

(2) uk,m(g)uk,m1(g) = uk,m+m1

(g);

(3) uk,m(g)uk1,m(gk) = ukk1,m(g);

(4) u1,m(g) = 1;

35

preliminaries

(5) u−1,m(g) = (−g)−m;

(6) uk,m(g)i = uk,mi(g), for a non-negative integer i;

(7) uk,m(g)−1 = uk1,m(gk), if kk1 ≡ 1 mod n;

(8) un−k,m(g) = uk,m(g)g−km provided (−1)m ≡ 1 mod n.

By (1), one can allow negative integers k with the obvious meaning and by

(6) and (7), an integral power of a Bass unit is a Bass unit.

Let N be a normal subgroup of G. Using equations (1) and (6) together

with the Chinese Remainder Theorem, it is easy to verify that some power of

a Bass unit in Z(G/N) is the natural image of a Bass unit in ZG.

By (4) and (5), it is clear that uk,m(g) is of finite order if k ≡ ±1 mod |g|.The opposite is proven in [PMS02, Proposition 8.1.12].

Proposition 1.12.3

Consider a finite group G, an element g ∈ G of order n and suppose that

k and m are positive integers for which km ≡ 1 mod n. The Bass unit

uk,m(g) has finite order if and only if k ≡ ±1 mod |g|.

1.13 bicyclic units

There are not many recipes known for constructing units in group rings. Be-

sides the famous Bass units, the construction of bicyclic units, introduced by

Jurgen Ritter and Sudarshan Sehgal in [RS91b], is also well known.

The bicyclic units of ZG are the elements of one of the following forms

βg,h = 1 + (1− g)h(1 + g + g2 + · · ·+ gn−1)

and

γg,h = 1 + (1 + g + g2 + · · ·+ gn−1)h(1− g),

where g, h ∈ G and g has order n. We denote by B2(G) the group generated

by the bicyclic units of ZG.

More generally, given a collection {e1, . . . , es} of idempotents of QG, one

defines generalized bicyclic units

βei,h = 1 + z2ei(1− ei)hei

36

1.13 bicyclic units

and

γei,h = 1 + z2eieih(1− ei),

with h ∈ G and where zei ∈ N is chosen of minimal value with respect to the

property that zeiei lies in ZG. One calls the group generated by the various

βei,h and γei,h the group of generalized bicyclic units and denotes it by GB2(G).

Note that, when using GB2(G), a collection of idempotents should be given at

least implicitly.

The idea of the following lemma comes from [JL93].

Lemma 1.13.1 (Jespers-Leal)

Let G be a finite group and {e1, . . . , em} a collection of primitive central

idempotents of QG. Consider the set {gei : g ∈ G, i ∈ {1, . . . ,m}} of

idempotents of QG. Then GB2(G) ⊆ B2(G).

Proof We will prove that γgei,h ∈ B2(G). Analogously, βgei,h ∈ B2(G) and

the result follows.

Take g, h ∈ G with |g| = n, and for i ∈ {1, . . . ,m}, take zi minimal such that

zingei ∈ ZG. Consider γgei,h = 1+z2i n

2geih(1−gei). Let α =∑n−1i=0 (n−i)gi ∈

ZG. It is clear that g(1− g) = 0. Easy computations verify that

α(1− g) = n(1− g) and γgei,h = 1 + z2i ngheiα(1− g).

Because of the equation

(1 + ngx(1− g))k(1 + ngy(1− g))l = 1 + ng(kx+ ly)(1− g),

it easily follows that

{1 + ngα(1− g) : α ∈ ZG} ⊆ B2(G).

Hence, γgei,h ∈ B2(G) and the proof is finished.

The following proposition is a reformulation of a result of Chapter 22 from

Sudarshan Sehgal’s book [Seh93].

Proposition 1.13.2

Assume G is a finite group and let U ⊆ U(ZG) be a subgroup of the unit

group of the integral group ring. Let QG =⊕n

i=1 QGei =⊕n

i=1Mni(Di)

be the Wedderburn decomposition of QG and let ZGei = Mni(Oi) denote

the orders in each simple component.

37

preliminaries

Then U is of finite index in U(ZG) if and only if both of the following

hold:

1. The natural image of U in K1(ZG) is of finite index;

2. For each i ∈ {1, . . . , n}, the group U contains a subgroup of finite

index in 1− ei + SLni(Oi).

By Theorem 1.12.1, one already knows that the natural image of B1(G) in

K1(ZG) is of finite index. This means that, in order to find generators for a

subgroup of finite index, one needs to look for units to enlarge B1(G) to a set

U such that the second condition is satisfied as well.

If QG does not contain exceptional components, then one knows that in

each Wedderburn component, the elementary matrices En(I), for any non-zero

ideal I of O, generate a subgroup of finite index in SLn(O), see Theorems 1.9.2

and 1.9.3. In this case, the usual method is to find units which behave similar

as the elementary matrices in each Wedderburn component of QG.

For several classes of finite groups G including nilpotent groups of odd order,

Jurgen Ritter and Sudarshan Sehgal [RS89, RS91b, RS91a] showed that B2(G)

fulfills the second condition and hence B1(G)∪B2(G) generates a subgroup of

finite index in U(ZG).

Let e be a primitive central idempotent of QG such that QGe = Mn(D) is

not exceptional and let O be a maximal order in D. Eric Jespers and Guil-

herme Leal proved in [JL93, Proposition 3.2] that if f is a non-central idempo-

tent in QGe, then the group generated by the generalized bicyclic units based

on f , does contain En(I) for some non-zero ideal I of O. This means that

B1(G)∪ GB2(G) will always satisfy the conditions of Proposition 1.13.2 when-

ever QG does not contain exceptional components and if one can construct a

non-central idempotent in each non-commutative Wedderburn component of

QG. The following result is a reformulation of [JL93, Theorem 3.3].

Proposition 1.13.3 (Jespers-Leal)

Let G be a finite group such that QG =⊕n

i=1 QGei does not contain

exceptional components. Let f1, . . . , fk be a collection of idempotents in

QG such that for each i ∈ {1, . . . , n} contributing to a non-commutative

component QGei, there is some index j(i) such that eifj(i) is non-central

in QGei. The set B1(G)∪ GB2(G) generates a subgroup of finite index in

U(ZG).

38

1.14 central units

If G has no non-abelian homomorphic image which is fixed point free, then

all projections Gei into the non-commutative components QGei, yield a non-

central idempotent “giei, for some gi ∈ G. The next result follows immediately

from Lemma 1.13.1 and Proposition 1.13.3, see [JL93, Corollary 4.1].

Corollary 1.13.4 (Jespers-Leal)

Let G be a finite group such that QG does not contain exceptional compo-

nents. If G has no non-abelian homomorphic image which is fixed point

free, then the set B1(G) ∪ B2(G) generates a subgroup of finite index in

U(ZG).

1.14 central units

Consider a finite group G and two of its elements g, h ∈ G. Recall that g and

h are said to be R-conjugate (respectively, Q-conjugate) if g is conjugate to

either h or h−1 (respectively, to hr for some integer r coprime with the order of

h) within G. This defines two equivalence relations on G and their equivalence

classes are called R-classes and Q-classes of G, respectively.

The following theorem is due to Samuil D. Berman and Ernst Witt and can

be found in [CR62, Theorem 42.8].

Theorem 1.14.1 (Berman-Witt)

The number of R- (respectively, Q-) classes of G coincides with the number

of simple components of the Wedderburn decomposition of RG (respecti-

vely, QG).

Let Z(U(ZG)) denote the group of central units in the integral group ring

ZG, for G a finite group. By Theorem 1.8.7, one knows that Z(U(ZG)) is

finitely generated. Moreover, inspired by Graham Higman, one deduces that

Z(U(ZG)) is equal to ±Z(G)× T , where T is a finitely generated free abelian

subgroup of Z(U(ZG)), see [PMS02, Corollary 7.3.3].

Using the theorem of Berman and Witt and Dirichlet’s Unit Theorem 1.8.8,

one can prove that the rank of Z(U(ZG)) for G a finite group is the difference

between the number of R-classes and Q-classes of G. From this, one deduces

the rank of Z(U(ZG)) in terms of the number of Wedderburn components, as

done, independently, by Jurgen Ritter and Sudarshan Sehgal [RS05, Theorem

2], and Raul Antonio Ferraz [Fer04, Theorem 3.6].

39

preliminaries

Theorem 1.14.2 (Ritter-Sehgal-Ferraz )

Let G be a finite group. Then Z(U(ZG)) is finitely generated and its

rank equals the difference between the number of simple components of

RG and the number of simple components of QG.

When G is an abelian group, the formula above simplifies as shown by

Graham Higman [Hig40]. The exact formula was given by Raymond G. Ayoub

and Christine Ayoub in [AA69].

Theorem 1.14.3 (Higman)

Let G be a finite abelian group. Then U(ZG) is finitely generated abelian

and has rank r = 1+k2+|G|−2c2 , where c is the number of cyclic subgroups

of G and k2 is the number of elements of G of order 2.

40

2W E D D E R B U R N D E C O M P O S I T I O N A N D I D E M P O T E N T S

A concrete realization of the Wedderburn decomposition of semisimple group

algebras by means of its central idempotents and orthogonal primitive idem-

potents is of importance for many topics. For example, Jespers and Leal have

shown one of the profits of idempotents to units of ZG in Proposition 1.13.3.

In this chapter, we extend the results from Theorem 1.7.9 and Proposi-

tion 1.7.13, due to Aurora Olivieri, Angel del Rıo and Juan Jacobo Simon, to

group algebras FG over number fields F . Secondly, we describe a complete

set of orthogonal primitive idempotents of QG and of finite semisimple group

algebras FG, when G is a finite strongly monomial group such that each strong

Shoda pair yields a trivial twisting τ .

2.1 the wedderburn decomposition of F G

In this section, we assume G to be a finite group and F to be a number field.

All character of G are assumed to be complex characters.

If G = 〈g〉 is cyclic of order k, then the irreducible characters are all linear

and are defined by the image of a generator of G. Therefore the set G∗ =

Irr(G) of irreducible characters of G is a group and the map ψ : Z/kZ → G∗

given by ψ(m)(g) = ζmk is a group homomorphism. The generators of G∗

are precisely the faithful representations of G. Let CF (G) denote the set of

orbits of the faithful characters of G under the action of Gal(F (ζk )/F ). Note

that for any faithful character χ of G, F (χ) = F (ζk ).

41

wedderburn decomposition and idempotents

Each automorphism σ ∈ Gal(F (ζk )/F ) is completely determined by its

image of ζk , and is given by σ(ζk ) = ζ tk , where t is an integer uniquely

determined modulo k. In this way, one gets the following morphisms

Gal(F (ζk )/F ) ��

//

'��

Gal(Q(ζk )/Q)

'��

Ik (F ) ��

// U (Z/kZ)

where we denote the image of Gal(F (ζk )/F ) in U (Z/kZ) by Ik (F ). In

this setting, the sets in CF (G) correspond one-to-one to the orbits under the

action of Ik (F ) on U (Z/kZ) by multiplication.

Let N � G be such that G/N is cyclic of order k and C ∈ CF (G/N ).

If χ ∈ C and tr = trF (ζk )/F denotes the field trace of the Galois extension

F (ζk )/F (i.e. tr(x) =∑

σ∈Gal(F (ζk )/F ) σ(x) for all x ∈ F (ζk )), then we

set

εC (G, N ) =1

|G|∑g∈G

tr(χ(gN ))g−1

=1

|G|∑g∈G

∑ψ∈C

ψ(gN )g−1

= [G : N ]−1“N ∑X∈G/N

∑ψ∈C

ψ(X )g−1X ,

where gX denotes a representative of X ∈ G/N . Note that εC (G, N ) does

not depend on the choice of χ ∈ C .

Let K � H ≤ G such that H/K is cyclic and C ∈ CF (H/K ). Then

eC (G, H, K ) denotes the sum of the different G-conjugates of εC (H, K ).

Proposition 2.1.1

If G is a finite abelian group of order n and F is a number field, then

the map (N,C) 7→ εC(G,N) is a bijection from the set of pairs (N,C)

with N � G, such that G/N is cyclic, and C ∈ CF (G/N) to the set of

primitive central idempotents of FG. Furthermore, for every N �G and

C ∈ CF (G/N), FGεC(G,N) ' F (ζk), where k = [G : N ].

Proof If e is a primitive central idempotent of FG, then there exists an

irreducible character ψ of G such that e = eF (ψ). Since G is abelian, ψ is

42


linear. Let N denote the kernel of ψ and let χ be the faithful character of G/N

given by χ(gN) = ψ(g). Then G/N is cyclic, say of order k, F (ψ) = F (ζk)

and the orbit of χ under the action of Gal(F (ζk)/F ) is an element of CF (G/N).

Furthermore

eF (ψ) =∑

σ∈Gal(F (ψ)/F )

σ · e(ψ)

=1

|G|∑

σ∈Gal(F (ψ)/F )

∑g∈G

σ(ψ(g))g−1

=1

|G|∑

σ∈Gal(F (ψ)/F )

∑g∈G

σ(χ(gN))g−1 (4)

=1

|G|∑g∈G

trF (ζk)/F (χ(gN))g−1

= εC(G,N).

This shows that the map is surjective and, since neF (ψ) = eF (ψ) for all n ∈ N ,

FGεC(G,N) = FGeF (ψ) ' F (ζk).

Assume now that εC1(G,N1) = εC2

(G,N2) with Ni�G and Ci ∈ CF (G/Ni).

Take χi ∈ Ci. Let πi : G→ G/Ni be the canonical projections and ψi = χi◦πi.Then ψi are irreducible characters of G. By (4), eF (ψ1) = εC1

(G,N1) =

εC2(G,N2) = eF (ψ2) and F (ψ1) = F (ψ2). If K = F (ψi), then there exists a

σ ∈ Gal(K/F ) such that ψ2 = σ ◦ ψ1 and hence N1 = kerψ1 = kerψ2 = N2.

Now let π−1 be a right inverse of π1 = π2. Then χ2 = ψ2◦π−1 = σ◦ψ1◦π−1 =

σ ◦ χ1 and hence C1 = C2. This shows that the map is injective.

Corollary 2.1.2

If G is a finite group with normal subgroup N such that G/N is cyclic of

order k and F is a number field, then εC(G,N) is a primitive central idem-

potent of FG and FGεC(G,N) ' F (ζk). Furthermore, if D ∈ CF (G/N),

then εC(G,N) = εD(G,N) if and only if C = D.

Proof The natural epimorphism G → G/N induces a ring isomorphism

ψ : FG“N ' F (G/N) (see (1) Section 1.6). Since εC(G,N) ∈ FG“N and

ψ(εC(G,N)) = εC(G/N, 1) is a primitive central idempotent of F (G/N) by

Proposition 2.1.1, also εC(G,N) is a primitive central idempotent in FG.

Moreover, again by Proposition 2.1.1, FGεC(G,N) ' F (G/N)εC(G/N, 1) 'F (ζk) and εC(G,N) = εD(G,N) if and only if εC(G/N, 1) = εD(G/N, 1) if

and only if C = D.

43


Let H be a subgroup of G, ψ a linear character of H and g ∈ G. Then ψg

denotes the character of Hg given by ψg(hg) = ψ(h). Since ker(ψg) = ker(ψ)g,

the map ψ 7→ ψg is a bijection between linear characters of H with kernel

K and linear characters of Hg with kernel Kg. This map induces a bijection

CF (H/K) → CF (Hg/Kg) : C 7→ Cg. In this sense, the following equation is

easy to verify:

εC(H,K)g = εCg (Hg,Kg). (5)

Now let K � H ≤ G be such that H/K is cyclic of order k. Then N =

NG(H) ∩NG(K) acts on CF (H/K) by the rule

{χi : i ∈ Ik(F )} · g = {(χg)i : i ∈ Ik(F )}.

Consider the stabilizer of a C ∈ CF (H/K), take χ ∈ C and fix a generator hK

of H/K,

StabN (C) = {g ∈ N : χg = χi for some i ∈ Ik(F )}= {g ∈ N : χ(g−1hgK) = χ(hiK) for some i ∈ Ik(F )}= {g ∈ N : g−1hgK = hiK for some i ∈ Ik(F )}.

Note that we used that χ is faithful in the third equality. Note that StabN (C)

is independent on the choice of generator of H/K. Indeed, let h1K and h2K

be generators such that h1K = hj2K. Then

StabN (C) = {g ∈ N : g−1h1gK = hi1K for some i ∈ Ik(F )}= {g ∈ N : (g−1h2gK)j = (hi2K)j for some i ∈ Ik(F )}⊇ {g ∈ N : g−1h2gK = hi2K for some i ∈ Ik(F )},

which gives an equality when reversing the role of h1 and h2. Hence StabN (C)

is independent on the choice of C ∈ CF (H/K) and hence it is the stabilizer

of any C ∈ CF (H/K) under the action of N . We denote this stabilizer by

EF (G,H/K). Note that Ik(Q) = U(Z/kZ) and EQ(G,H/K) = N .

By Proposition 1.7.8, all irreducible characters of a monomial group G are

associated with Shoda pairs of G. The next theorem provides a description of

the primitive central idempotent given by a Shoda pair (H,K) of a group G

as a multiple of eC(G,H,K).

44


Theorem 2.1.3

Let G be a finite group, (H,K) a Shoda pair of G and F be a number

field. Let χ be a linear character of H with kernel K and C be the orbit

of χ under the action of Gal(F (χ)/F ). Then χG is irreducible and the

primitive central idempotent of FG associated to χG is

eF (χG) =[CenG(εC(H,K)) : H]

[F (χ) : F (χG)]eC(G,H,K).

Proof Let e = e(χ). Let Gal(F (χ)/F ) = {σ1, . . . , σn} and T = {g1, . . . , gm}be a right transversal of H in G. Then

e(χG) =1

|G|∑g∈G

χG(1)χG(g−1)g

=1

|G|∑g∈G

|G||H|

χ(1)

(m∑i=1

χ◦(gig−1g−1

i )

)g

=1

|H|

m∑i=1

∑h∈H

χ(h−1)g−1i hgi

=m∑i=1

e · gi.

Consider the table

σ1 · e · g1 σ1 · e · g2 · · · σ1 · e · gm σ1 · e(χG)

σ2 · e · g1 σ2 · e · g2 · · · σ2 · e · gm σ2 · e(χG)

· · · · · · · · · · · · · · ·σn · e · g1 σn · e · g2 · · · σn · e · gm σn · e(χG)

εC(H,K) · g1 εC(H,K) · g2 · · · εC(H,K) · gm ∗

We can compute the total sum ∗ by adding the elements of the last column

or the elements of the last row:

∗ =n∑i=1

σi · e(χG) =m∑j=1

εC(H,K) · gj . (6)

45


In the first sum of (6) the elements to add are the elements of the orbit of

e(χG) under the action of Gal(F (χ)/F ), each of them repeated [F (χ) : F (χG)]

times. Using the formula of Yamada (2) from Section 1.7, one has

∗ = [F (χ) : F (χG)]eF (χG). (7)

Similarly the second sum of (6) adds up the elements of the G-orbit of εC(H,K)

by (5), each of them repeated [CenG(εC(H,K)) : H] times. Therefore

∗ = [CenG(εC(H,K)) : H]eC(G,H,K). (8)

The theorem follows by comparing (7) and (8).

So for each Shoda pair (H,K) of G, number field F and each C ∈ CF (H/K),

there exists a unique α ∈ Q such that αeC(G,H,K) is a primitive central

idempotent of FG.

Next, we will investigate the case when α = 1. This happens when (H,K)

is a strong Shoda pair of G.

Lemma 2.1.4

Let F be a number field. Let K �H ≤ G be such that H/K is cyclic of

order k and C ∈ CF (H/K).

1. For every g ∈ G, the following statements are equivalent:

a) g ∈ K;

b) gεC(H,K) = εC(H,K);

c) gεC(H,K) = εC(H,K).

2. If H �NG(K), then CenG(εC(H,K)) = EF (G,H/K).

Proof 1. The fact that 1a) implies 1b) follows from the easy observation

that g“K = “K when g ∈ K. The equivalence between 1b) and 1c) fol-

lows by comparing the coefficients. It remains to prove that g ∈ K when

gεC(H,K) = εC(H,K). Assume that gεC(H,K) = εC(H,K). Because of

Corollary 2.1.2, εC(H,K) is a primitive central idempotent in FH and hence

46


non-zero. Therefore, the support of gεC(H,K) is a non-empty set in H and

g ∈ H. Hence we can write g = xht for some x ∈ K and 0 ≤ t ≤ k. Now

k−1“K k−1∑i=0

∑ψ∈C

ψ(hiK)ht−i = gεC(H,K)

= εC(H,K)

= k−1“K k−1∑i=0

∑ψ∈C

ψ(hiK)h−i

and hence

trF (ζk)/F ((χ(htK)− 1)χ(hiK)) =∑ψ∈C

(ψ(htK)− 1)ψ(hiK) = 0

for every 0 ≤ i ≤ k−1 and some χ ∈ C. Since χ is faithful, its image generates

F (χ) = F (ζk) as F -vector space. Since trF (ζk)/F : F (ζk)→ F is F -linear and

surjective, we deduce that χ(htK) = 1 and hence k divides t. Therefore t = 0

and g ∈ K.

2. When H � NG(K), clearly EF (G,H/K) ⊆ NG(H) ∩ NG(K) ⊆ NG(K).

Furthermore, if g ∈ CenG(εC(H,K)), then for each x ∈ K

g−1xgεC(H,K) = g−1xεC(H,K)g = εC(H,K),

by 1. Therefore g−1xg ∈ K and CenG(εC(H,K)) ⊆ NG(K). Take g ∈ NG(K).

By Corollary 2.1.2, εC(H,K) and εCg (H,K) are two primitive central idempo-

tents of FH and they are equal if and only if C = Cg (i.e. if g ∈ EF (G,H/K)).

By (5), εC(H,K)g = εCg (H,K). Hence g ∈ CenG(εC(H,K)) if and only if

εC(H,K)g = εC(H,K), if and only if g ∈ EF (G,H/K).

Clearly, elements of QG can also be seen as elements in FG. The following

lemma tells how ε(G,N) and e(G,H,K) can be written as a sum of elements

in FG.

Lemma 2.1.5

Let F be a number field.

1. Let N �G be such that G/N is cyclic, then

ε(G,N) =∑

C∈CF (G/N)

εC(G,N).

47


2. Let K �H �NG(K) be such that H/K is cyclic and let R be a set

of representatives of the action of NG(K) on CF (H/K). Then

e(G,H,K) =∑C∈R

eC(G,H,K).

Proof 1. For every C ∈ CF (H/K), both ε(G,N) and εC(G,N) belong to

FG“N . By factoring out N and using the isomorphism FG“N ' F (G/N),

we may assume without loss of generality that N = 1 and G is cyclic. By

Proposition 2.1.1, every primitive central idempotent of FG is of the form

εC(G,H) with H�G and C ∈ CF (G/H). Therefore ε(G, 1) is the sum of some

elements εC(G,H) and it is enough to prove that if H �G and C ∈ CF (G/H),

then ε(G, 1)εC(G,H) 6= 0 if and only if H = 1.

If C ∈ CF (G) and 1 6= x ∈ G, then (1 − x)εC(G, 1) 6= 0 by Lemma 2.1.4.

Since εC(G, 1) is a primitive central idempotent, we have that εC(G, 1) =

(1 − x)εC(G, 1). Since ε(G, 1) is the product of elements of the form 1 − xwith 1 6= x ∈ G, it follows that ε(G, 1)εC(G, 1) = εC(G, 1) 6= 0. Conversely, if

1 6= H ≤ G, then there exists a h ∈ H such that 〈h〉 is a minimal non-trivial

subgroup of G. Hence ε(G, 1)εC(G,H) = ε(G, 1)(1 − h)εC(G,H) = 0, by

Lemma 2.1.4. This finishes part 1 of the proof.

2. Let N = NG(H) ∩ NG(K) = NG(K), E = EF (G,H/K), TN be a

right transversal of N in G, TE be a right transversal of E in N . Then

{hg : h ∈ TE , g ∈ TN} is a right transversal of E in G. By Proposition 1.7.12,

we know that N = CenG(ε(H,K)). Hence e(G,H,K) =∑g∈TN ε(H,K)g.

Clearly, CF (H/K) is the disjoint union of the sets {Ct : t ∈ TE} for C running

on R. Therefore,

e(G,H,K) =∑g∈TN

ε(H,K)g

=∑g∈TN

∑C∈CF (H/K)

εC(H,K)g

=∑g∈TN

∑C∈R

∑h∈TE

εCh(H,K)g

=∑C∈R

∑g∈TN

∑h∈TE

εC(H,K)hg

=∑C∈R

eC(G,H,K).

48


In the main theorem of this section we describe the simple components

of the group algebra FG provided by strong Shoda pairs. We show that a

strong Shoda pair (H,K) of G that determines a primitive central idempo-

tent e(G,H,K) in QG, will also determine a primitive central idempotent

eC(G,H,K) in FG for C ∈ CF (G/N).

Theorem 2.1.6

Let G be a finite group and F be a number field.

1. Let (H,K) be a strong Shoda pair of G and C ∈ CF (H/K). Let

[H : K] = k, yK a generator of H/K and E = EF (G,H/K). Then

eC(G,H,K) is a primitive central idempotent of FG and

FGeC(G,H,K) 'M[G:E] (F (ζk) ∗στ E/H) ,

where σ and τ are defined as follows. Let ψ : E/H → E/K be a left

inverse of the projection E/K → E/H. Then

σgH(ζk) = ζik, if yKψ(gH) = yiK,

τ(gH, g′H) = ζjk, if ψ(gg′H)−1ψ(gH)ψ(g′H) = yjK,

for gH, g′H ∈ E/H and integers i and j.

2. Let X be a set of strong Shoda pairs of G. If every primitive central

idempotent of QG is of the form e(G,H,K) for (H,K) ∈ X, then

every primitive central idempotent of FG is of the form eC(G,H,K)

for (H,K) ∈ X and C ∈ CF (H/K).

Proof 1. By Lemma 2.1.4, E = CenG(εC(H,K)) and by Proposition 1.7.12,

CenG(ε(H,K)) = NG(K). Let T be a right transversal of E in G, then

eC(G,H,K) =∑g∈T εC(H,K)g.

In order to prove that eC(G,H,K) is an idempotent, it is enough to show

that the G-conjugates of εC(H,K) are orthogonal. For this we show that, if

g ∈ G \ E, then εC(H,K)εC(H,K)g = 0. By Lemma 2.1.5,

εC(H,K)εC(H,K)g = εC(H,K)ε(H,K)ε(H,K)gεC(H,K)g,

49


which is zero by the definition of a strong Shoda pair whenever g /∈ NG(K).

If g ∈ NG(K) \E, then εC(H,K)g = εCg (H,K) 6= εC(H,K) are two different

primitive central idempotents of FH by Corollary 2.1.2. Hence

εC(H,K)εC(H,K)g = 0.

By Corollary 2.1.2, FHεC(H,K) ' F (ζk). Also, by Lemma 2.1.4,

FEεC(H,K) = FHεC(H,K) ∗στ E/H

is a crossed product with homogeneous basis ψ(E/H), where ψ : E/H → E/K

is a left inverse of the projection E/K → E/H. The action σ and twisting τ

are given by

σ : E/H → Aut(FHεC(H,K))

gH 7→ (αεC(H,K) 7→ ψ(gH)−1αεC(H,K)ψ(gH)),

τ : E/H × E/H → U(FHεC(H,K))

(gH, g′H) 7→ ψ(gg′H)−1ψ(gH)ψ(g′H).

Clearly, the isomorphism FHεC(H,K) ' F (ζk) extends to an E/H-graded

isomorphism

FEεC(H,K) = FHεC(H,K) ∗στ E/H ' F (ζk) ∗σ′

τ ′ E/H.

Since H/K is maximal abelian in N/K and hence also in E/K, the action σ′

is faithful and FEεC(H,K) is a simple algebra (Theorem 1.5.1).

If g ∈ G, then the map x 7→ xg defines an isomorphism between the FG-

modules FGεC(H,K) and FGεC(H,K)g. Therefore,

FGeC(G,H,K) =⊕t∈T

FGεC(H,K)t ' (FGεC(H,K))[G:E],

as FG-modules. Moreover,

εC(H,K)FGεC(H,K) =⊕t∈T

εC(H,K)FEtεC(H,K)

=⊕t∈T

FEεC(H,K)tεC(H,K)

= FEεC(H,K),

50


because εC(H,K) is central in FE and εC(H,K)tεC(H,K) = 0 for all t ∈G \ E. Thus

FGeC(G,H,K) ' EndFG(FGeC(G,H,K))

' M[G:E](EndFG(FGεC(H,K)))

' M[G:E](εC(H,K)FGεC(H,K))

' M[G:E](FEεC(H,K))

' M[G:E](F (ζk) ∗σ′

τ ′ E/H).

2. By assumption there is a set Y ⊆ X such that {e(G,H,K) : (H,K) ∈ Y }is the set of primitive central idempotents of QG. Hence, by Lemma 2.1.5,

1 =∑

(H,K)∈Y

e(G,H,K) =∑

(H,K)∈Y

∑C∈R(H,K)

eC(G,H,K),

for R(H,K) a set of representatives of the action of NG(K) on CF (H/K).

Furthermore, eC(G,H,K) are primitive central idempotents of FG by part

1. Hence {eC(G,H,K) : (H,K) ∈ Y,C ∈ R(H,K)} is a complete and non-

redundant set of primitive central idempotents of FG.

Applying Proposition 1.7.14, we get the following result.

Corollary 2.1.7

If G is a finite strongly monomial group and F is a number field, then

every primitive central idempotent of FG is of the form eC(G,H,K) for

a strong Shoda pair (H,K) of G and C ∈ CF (H/K). Furthermore, for

every strong Shoda pair (H,K) of G and every C ∈ CF (H/K),

FGeC(G,H,K) 'M[G:E]

(F(ζ[H:K]

)∗στ E/H

),

where σ and τ are defined as in Theorem 2.1.6 and E = EF (G,H/K).

Applying Theorem 1.7.16, we get the following result.

51


Corollary 2.1.8

If G is a finite metabelian group, A a maximal abelian subgroup of G

containing G′ and F a number field. Then every primitive central idem-

potent of FG is of the form eC(G,H,K) for a pair (H,K) of subgroups

of G satisfying the following conditions:

1. H is a maximal element in {B ≤ G : A ≤ B and B′ ≤ K ≤ B};

2. H/K is cyclic;

and C ∈ CF (H/K).

Example 2.1.9 Consider the nilpotent (and thus strongly monomial) group

G = C3 ×Q8 with presentation⟨a, x, y : a3 = 1, x4 = 1, y2 = x2, y−1xy = x−1, ax = xa, ay = ya

⟩.

Using the GAP-package wedderga [BHK+14], one can compute the following

Wedderburn decompositions:

QG = 4Q⊕ 4Q(ζ3)⊕Å−1,−1

Q

ã⊕M2(Q(ζ3)),

Q(ζ3)G = 12Q(ζ3)⊕ 3M2(Q(ζ3)).

The corresponding strong Shoda pairs are

(G,G), (G, 〈a, x〉), (G, 〈a, y〉), (G,⟨a, xy−1

⟩), (G, 〈x, y〉),

(G, 〈x〉), (G, 〈y〉), (G,⟨xy−1

⟩), (〈a, x〉 , 〈a〉) and (〈a, x〉 , 1). (9)

The last two pairs contribute to the non-commutative components of QGand Q(ζ3)G.

Let H = 〈a, x〉 and K = 1. The strong Shoda pair (H,K) realizes the

component M2(Q(ζ3)) of QG and hence also the components

Q(ζ3)⊗Q M2(Q(ζ3)) = 2M2(Q(ζ3))

of Q(ζ3)G. We will show how to construct the corresponding idempotents.

Let H and K be as before, then [H : K] = 12 and H/K ' C12. The

unit group of Z/12Z equals {1, 5, 7, 11} and hence I12(Q) = {1, 5, 7, 11}. Con-

sider now the Galois group Gal(Q(ζ12)/Q(ζ3)). This group contains precisely

the automorphisms determined by ζ12 7→ ζi12 with i ∈ {1, 7}. Therefore,

52


I12(Q(ζ3)) = {1, 7}. From now on, we denote the map ζ12 7→ ζi12 by the

integer i. We compute

CQ(H/K) = {{1, 5, 7, 11}} and CQ(ζ3)(H/K) = {{1, 7}, {5, 11}}.

Since CQ(H/K) contains only one element, the pair (H,K) leads to one

primitive central idempotent of QG. Take C = {1, 5, 7, 11} ∈ CQ(H/K). Then,

εC(H,K) =1

12

∑h∈H

∑ψ∈C

ψ(h)h−1

=1

12

12∑i=1

∑ψ∈C

ψ(xa)i(xa)−i

=1

12

12∑i=1

(ζi12 + ζ5i12 + ζ7i

12 + ζ11i12 )(xa)−i.

Since CQ(ζ3)(H/K) contains two elements, the pair (H,K) leads to two prim-

itive central idempotents of Q(ζ3)G. Take C1 = {1, 7} and C2 = {5, 11} ∈CQ(ζ3)(H/K), then

εC1(H,K) =

1

12

12∑i=1

(ζi12 + ζ7i12)(xa)−i,

εC2(H,K) =

1

12

12∑i=1

(ζ5i12 + ζ11i

12 )(xa)−i, and

εC(H,K) = εC1(H,K) + εC2

(H,K).

Since y−1axy = ax−1 = (ax)7 and 7 ∈ I12(Q(ζ3)), both EQ(G,H/K) and

EQ(ζ3)(G,H/K) equal G. Therefore eC(G,H,K) = εC(H,K), eC1(G,H,K) =

εC1(H,K) and eC2(G,H,K) = εC2(H,K).

Consider the simple components QGeC(G,H,K), Q(ζ3)eC1(G,H,K) and

Q(ζ3)eC2(G,H,K). By Theorem 2.1.6, they are all isomorphic to the crossed

product Q(ζ12) ∗ 〈yH〉 =∑2i=1 Q(ζ12)uyi , where u2

y = ζ612 = −1 since y2 =

x2 = (ax)6 and u−1y ζ12uy = ζ7

12 since y−1axy = ax−1 = (ax)7. We can rewrite

Q(ζ12) ∗ 〈yH〉 ' Q(ζ3)(i, u : i2 = −1, u2 = −1, u−1iu = i7 = −i)

'Å−1,−1

Q(ζ3)

ã.

53


Since −ζ23 − (ζ2

3 )2 = 12 is satisfied in Q(ζ3), this quaternion algebra splits by

Proposition 1.2.1 and equals M2(Q(ζ3)).

Remark 2.1.10 There is a strong correspondence between the simple com-

ponents in semisimple finite group algebras and simple components in group

algebras over number fields. Let Fq be the finite field of order q = pn then

Fq(ζk) = Fqo for an integer k coprime to p and o = ok(q). The Galois group

Gal(Fq(ζk)/Fq) is cyclic of order ok(q) and can be seen as a subgroup of

U(Z/kZ). For a cyclic group G, the elements of CF(G) are also referred to

as the q-cyclotomic classes of G.

Hereby, we find back the following result from [BdR07, Theorem 7].

Corollary 2.1.11 (Broche-del Rıo)

Let G be a finite group and F a finite field of order q such that FG is

semisimple. Let (H,K) be a strong Shoda pair of G and C ∈ CF(H/K).

Then eC(G,H,K) is a primitive central idempotent of FG and

FGeC(G,H,K) 'M[G:H](Fqo/[E:H]),

where E = EF(G,H/K) and o is the multiplicative order of q modulo

[H : K].

Remark 2.1.12 From [BdR07, Theorem 7], one also knows that there is a

strong relation between the primitive central idempotents in a rational group

algebra QG and the primitive central idempotents in a finite semisimple group

algebra FG which are realized by the strong Shoda pairs of G. More precisely,

if X is a set of strong Shoda pairs of G and every primitive central idempo-

tent of QG is of the form e(G,H,K) for (H,K) ∈ X, then every primitive

central idempotent of FG is of the form eC(G,H,K) for (H,K) ∈ X and

C ∈ CF(H/K).

Corollary 2.1.13

If G is a strongly monomial group and F is a finite field of order q such that

FG is semisimple, then every primitive central idempotent of FG is of the

form eC(G,H,K) for (H,K) a strong Shoda pair of G and C ∈ CF(H/K).

Example 2.1.14 Consider again the group G = C3×Q8 from Example 2.1.9.

All primitive central idempotents of F5G are realized by the strong Shoda

pairs from (9). Consider H = 〈a, x〉 and K = 1, then [H : K] = 12 and

54

2.2 primitive idempotents of QG

F5(ζ12) = F5o12(5) = F25. The Galois group of the extension F25/F5 is gen-

erated by the Frobenius automorphism x 7→ x5. Hence I12(F5) = {1, 5}.Since y−1axy = (ax)7 and 7 /∈ I12(F5), EF5

(G,H/K) = H. Furthermore,

CF5(H/K) = {{1, 5}, {7, 11}}. Let C1 = {1, 5} and C2 = {7, 11}, then

eC1(G,H,K) = εC1(H,K) + y−1εC1(H,K)y

= εC1(H,K) + εC2(H,K)

= eC2(G,H,K).

Also F5GeC1(G,H,K) 'M2(F25).


As mentioned in Example 1.7.20, it is known that the construction of the

orthogonal idempotents in Theorem 1.7.19 cannot be extended to, for example,

arbitrary finite metacyclic groups. We present an alternative construction to

describe a complete set of orthogonal primitive idempotents for a class of finite

strongly monomial groups containing the finite metacyclic groups Cqm o1Cpn .

In this section we will focus on simple components of QG of a finite group

G which are determined by a strong Shoda pair (H, K ) such that the twist-

ing τ (gH, g ′H ) = 1 for all g , g ′ ∈ NG (K ) (with notation as in Proposi-

tion 1.7.13). For such a component, we describe a complete set of orthogonal

primitive idempotents (and a complete set of matrix units). This construction

is based on the isomorphism of Theorem 1.5.2 on classical crossed products

with trivial twisting. Such a description, together with the description of the

primitive central idempotent e = e(G, H, K ) determining the simple compo-

nent, yields a complete set of irreducible modules.

Before we do so, we need a basis of Q(ζ [H :K ] )/Q(ζ [H :K ] )NG (K )/H of the

form {wx : x ∈ NG (K )/H } with w ∈ Q(ζ [H :K ] ). That such a basis exists

follows from the Normal Basis Theorem 1.3.1.

Theorem 2.2.1

Let (H,K) be a strong Shoda pair of a finite group G such that thetwisting τ(gH, g′H) = 1 for all g, g′ ∈ NG(K). Let ε = ε(H,K) ande = e(G,H,K). Let F denote the fixed subfield of QHε under the na-tural action of NG(K)/H and [NG(K) : H] = n. Let w be a normalelement of QHε/F and B the normal basis determined by w. Let ψ bethe isomorphism between QNG(K)ε and the matrix algebra Mn(F ) with

55


respect to the basis B as stated in Theorem 1.5.2. Let P,A ∈ Mn(F ) bedefined as follows:

P =

1 1 1 · · · 1 1

1 −1 0 · · · 0 0

1 0 −1 · · · 0 0...

......

. . ....

...

1 0 0 · · · −1 0

1 0 0 · · · 0 −1

and A =

0 0 · · · 0 1

1 0 · · · 0 0

0 1 · · · 0 0...

.... . .

......

0 0 · · · 0 0

0 0 · · · 1 0

.

Then

{xT1εx−1 : x ∈ T2 〈xe〉}

is a complete set of orthogonal primitive idempotents of QGe where we set

xe = ψ−1(PAP−1), T1 is a transversal of H in NG(K) and T2 is a right

transversal of NG(K) in G. By T1 we denote the element 1|T1|

∑t∈T1

t in

QG.

Proof Consider the simple component

QGe 'M[G:N ](QNε) 'M[G:N ](QHε/F, 1)

of QG with N = NG(K). Without loss of generality we may assume that K is

normal in G and hence N = G. Indeed, if we obtain a complete set of orthog-

onal primitive idempotents of QNε, then the conjugates by the transversal T2

of N in G will give a complete set of orthogonal primitive idempotents of QGesince e =

∑t∈T2

εt and different εt’s are orthogonal.

From now on we assume that N = G and e = ε. Then B = {wgH : g ∈ T1}.Because (H,K) is a strong Shoda pair of G, the group H/K is a maximal

abelian subgroup of NG(K)/K. Since G/H acts on QHe via the induced

conjugation action on H/K it easily is seen that the action of G/H on B is

regular (i.e. transitive and free). Hence it is readily verified that for each

g ∈ T1, ψ(ge) is a permutation matrix, and

ψ(T1e) =1

n

1 1 · · · 1 1

1 1 · · · 1 1

1 1 · · · 1 1...

.... . .

......

1 1 · · · 1 1

1 1 · · · 1 1

.

56


Clearly ψ(T1e) has eigenvalues 1 and 0, with respective eigenspaces

V1 = vect{(1, 1, . . . , 1)}

and

V0 = vect{(1,−1, 0, . . . , 0), (1, 0,−1, . . . , 0), . . . , (1, 0, 0, . . . ,−1)},

where vect(S) denotes the vector space generated by the set S. Hence

ψ(T1e) = PE11P−1,

where we denote by Eij ∈Mn(F ) the elementary matrices whose entries are all

0 except in the (i, j)-spot, where it is 1. One knows that {E11, E22, . . . , Enn}and hence also

{ψ(T1e) = PE11P−1, PE22P

−1, . . . , PEnnP−1}

forms a complete set of orthogonal primitive idempotents of Mn(F ). Let y =

ψ(xe) = PAP−1. As

E22 = AE11A−1, . . . , Enn = An−1E11A

−n+1

we obtain that

{ψ(T1e), yψ(T1e)y−1, . . . , yn−1ψ(T1e)y

−n+1}

forms a complete set of orthogonal primitive idempotents of Mn(F ). Hence,

applying ψ−1 gives us a complete set of orthogonal primitive idempotents of

QGe.

Next we will describe a complete set of matrix units in a simple component

QGe(G,H,K) for a strong Shoda pair (H,K) of a finite group G.

Corollary 2.2.2

Let (H,K) be a strong Shoda pair of a finite group G such that the twist-

ing τ(gH, g′H) = 1 for all g, g′ ∈ N . We use the notation of Theorem 2.2.1

and for every x, x′ ∈ T2 〈xe〉, let

Exx′ = xT1εx′−1.

Then {Exx′ : x, x′ ∈ T2 〈xe〉} is a complete set of matrix units in QGe, i.e.

e =∑x∈T2〈xe〉Exx and ExyEzw = δyzExw, for every x, y, z, w ∈ T2 〈xe〉.

Moreover ExxQGExx ' F , where F is the fixed subfield of QHε under

the natural action of NG(K)/H.

57


Proof This follows at once from Theorem 2.2.1 and the fact that QGe 'M[G:H](F ).

In order to obtain an internal description within the group algebra QG,

one would like to write the element xe = ψ−1(PAP−1) of Theorem 2.2.1 in

terms of group ring elements of QG. It might be a hard problem to obtain a

generic formula. One of the reasons is that we first need to describe a normal

basis of Q(ζ[H:K])/Q(ζ[H:K])N/H . In general this is difficult to do. However,

one can find some partial results in the literature. For example Dirk Hachen-

berger [Hac00] studied normal bases for cyclotomic fields Q(ζqm) with q an

odd prime number. Once this obstacle is overcome one can determine xe as

follows. Denote by ∆ : CN → C the trace map∑g∈N agg 7→ a1. It is easy to

see and well known that ∆(α) = 1|N |χreg(α) = 1

|N |∑χ∈Irr(N) χ(1)χ(α), where

we denote by χreg the regular character of N and by Irr(N) the set of irre-

ducible complex characters of N . It follows that xe =∑g∈N ∆(xeg

−1)g =1|N |∑g∈N

∑χ∈Irr(N) χ(1)χ(xeg

−1)g. Because ψ can be seen as the representa-

tion induced to N by a linear character of H with kernel K, ψ is an irreducible

complex representation of N . As we know that xe belongs to a simple compo-

nent of QN , namely the only one on which ψ does not vanish, and the primitive

central idempotent of QN of such component is the sum of the primitive central

idempotents of CN associated to the irreducible characters of the form σ◦T ◦ψ,

with σ ∈ Gal(F/Q) and T is the map associating a matrix with its trace. We

deduce that χ(xeg−1) vanishes in all the irreducible characters different from

σ ◦T ◦ψ. Thus xe = 1|N |∑g∈N

∑σ∈Gal(F/Q)(σ ◦T ◦ψ)(1)(σ ◦T ◦ψ)(xeg

−1)g =1|H|∑g∈N trF/Q(T (PAP−1ψ(g−1)))g.

We now show that we can sometimes also overcome the difficulties above

using only basic linear algebra for the group C7o1C3, where the method from

Theorem 1.7.19 failed.

Example 2.2.3 Let G = C7 o1 C3 =⟨a, b : a7 = 1 = b3, ab = a2

⟩. Then a

GAP-computation shows that

QG = Q⊕Q(ζ3)⊕M3(Q(ζ7 + ζ27 + ζ4

7 )).

Consider the strong Shoda pair (〈a〉 , 1) that contributes to the unique non-

commutative simple component M3(Q(ζ7 +ζ27 +ζ4

7 )). The associated primitive

central idempotent is e = e(G, 〈a〉 , 1) = ε(〈a〉 , 1) and

M3(Q(ζ7 + ζ27 + ζ4

7 )) ' QGe ' Q 〈a〉 e ∗ 〈b〉

58


with trivial twisting. Consider the algebra isomorphism

ψ : Q 〈a〉 e ∗ 〈b〉 'M3(Q(ae+ a2e+ a4e))

with respect to B = {ae, a2e, a4e}, a normal basis of Q(ae) over its subfield

Q(ae + a2e + a4e). Now we have A = ψ(be) and in order to describe xe =

ψ−1(PAP−1) in terms of elements of QG, it is sufficient to write ψ−1(P ) in

terms of group ring elements. Write ψ−1(P ) = α0+α1b+α2b2 with αi ∈ Q 〈a〉 e

and solve the system of equations:(α′0 + α′1 ◦ b+ α′2 ◦ b2)(ae) = (a+ a2 + a4)e

(α′0 + α′1 ◦ b+ α′2 ◦ b2)(a2e) = (a− a2)e

(α′0 + α′1 ◦ b+ α′2 ◦ b2)(a4e) = (a− a4)e.

This is done by writing each αi = (xi,0+xi,1a+xi,2a2+xi,3a

3+xi,4a4+xi,5a

5)e

with xi,j ∈ Q and using the equality (1+a+a2 +a3 +a4 +a5 +a6)e = 0. This

leads to a system of 18 linear equations in 18 variables. It can be verified that

ψ−1(P ) =

Å−4

7− 1

14a− 1

2a2 − 5

14a3 − 1

7a4 − 5

14a5

ãe

+

Å2

7− 11

14a− 3

14a2 − 1

2a3 − 1

7a4 − 9

14a5

ãbe

+

Å2

7− 9

14a− 11

14a2 − 9

14a3 +

2

7a4 − 1

2a5

ãb2e.

By Theorem 2.2.1, the set {be, x−1e bexe, x

−2e bex2

e} is a complete set of or-

thogonal primitive idempotents in QGe.This method of Theorem 2.2.1 yields a detailed description of a complete

set of orthogonal primitive idempotents of QG when G is a finite strongly

monomial group such that there exists a complete and non-redundant set of

strong Shoda pairs (H,K) satisfying τ(gH, g′H) = 1 for all g, g′ ∈ NG(K).

However, even when the group is not strongly monomial or some strong Shoda

pairs yield a non-trivial twisting, our description of primitive idempotents can

still be used in the components determined by a strong Shoda pair with trivial

twisting.

Nevertheless, it is crucial that the twistings appearing in the simple compo-

nents are trivial in order to make use of Theorem 1.5.2. The following example

shows that our methods cannot be extended to, for example, Cq o Cp2 with

non-faithful action.

59


Example 2.2.4 Consider the group

G = C19 o3 C9 = 〈a, b : a19 = b9 = 1, ab = a7〉

and the strong Shoda pair (⟨a, b3

⟩, 1). Let e be the associated primitive central

idempotent e = e(G,⟨a, b3

⟩, 1) = ε(

⟨a, b3

⟩, 1). The elements 1, b, b2 are coset

representatives for⟨a, b3

⟩in G. Since b2

⟨a, b3

⟩b2⟨a, b3

⟩= b

⟨a, b3

⟩and b3 =

(ab3)19, we get that τb2〈a,b3〉,b2〈a,b3〉 = ζ1957 6= 1. Hence the twisting is not trivial.

We show now that our method applies to all metacyclic groups of the form

Cqm o1 Cpn , for p and q different prime numbers. Recall that this notation

means that Cpn acts faithfully on Cqm .

Corollary 2.2.5

Let p and q be different prime numbers, m and n positive integers and

G = 〈a〉o1 〈b〉 with |a| = qm and |b| = pn. Each non-commutative simple

component of QG is realized as QGε(〈a〉 ,Kj) for a strong Shoda pair

(〈a〉 ,Kj) fore some 1 ≤ j ≤ m and where Kj =¨aqj∂.

Furthermore, let Fj be the center of QGε(〈a〉 ,Kj), fix a normal el-

ement wj of Q(ζqj )/Fj and let Bj be the normal basis determined by

wj . Let ψj : QGε(〈a〉 ,Kj)→Mpn(Fj) be the isomorphism given by The-

orem 1.5.2 with respect to Bj . Then,

{xhj bx−kj : 1 ≤ h, k ≤ pn}

is a complete set of matrix units of QGε(〈a〉 ,Kj), where

xj = ψj−1(P )bε(〈a〉 ,Kj)ψj

−1(P )−1.

Proof The primitive central idempotents and simple components of QG are

described in Corollary 1.7.18. Therefore, the first statement follows immedi-

ately.

Let Kj =¨aqj∂

and consider the simple component

QGε (〈a〉 ,Kj) ' Q(ζqj)∗ Cpn .

It is easy to verify that the twisting is trivial. Let Fj = Q(ζqj)Cpn , the fixed

field of Q(ζqj)

by the action of Cpn . Let Bj be the normal basis determined by

a normal element wj of Q(ζqj )/Fj . Let ψj : QGε(〈a〉 ,Kj)→Mpn(Fj) be the

60

2.3 primitive idempotents of FG

isomorphism given by Theorem 1.5.2 with respect to Bj . Then ψj(bε(〈a〉 ,Kj))

is the permutation matrix A of Theorem 2.2.1 and 〈b〉 is a transversal of 〈a〉in G. The result follows now from Corollary 2.2.2.

As we have shown, the groups of the form Cqmo1Cpn do satisfy the condition

of a trivial twisting. However not all groups satisfying this condition on the

twistings are metacyclic, for example the symmetric group S4 and the alternat-

ing group A4 of degree 4 have a trivial twisting in all Wedderburn components

of their rational group rings and are not metacyclic (and not nilpotent). Triv-

ially all abelian groups are included and it is also easy to prove that for all

dihedral groups D2n =⟨a, b : an = b2 = 1, ab = a−1

⟩there exists a complete

and non-redundant set of strong Shoda pairs with trivial twisting since the

group action involved has order 2 and hence is faithful. On the other hand for

quaternion groups Q4n =⟨x, y : x2n = y4 = 1, xn = y2, xy = x−1

⟩, one can

verify that the strong Shoda pair (〈x〉 , 1) yields a non-trivial twisting.


In this section, we construct a complete set of orthogonal primitive idempo-

tents in some semisimple finite group algebras FG.

Apart from the ring theoretical interest, primitive idempotents can be used

to construct linear codes.

Let F be a finite field. A linear code over F of length n and rank k is a linear

subspace C with dimension k of the vector space Fn. The standard basis of

Fn is denoted by E = {e1 , . . . , en}. The vectors in C are called codewords,

the size of a code is the number of codewords and equals |F|k .

If G is a group of order n and C ⊆ Fn is a linear code, then one says that C

is a left G-code (respectively a G-code) if there is a bijection φ : E → G such

that the linear extension of φ to an isomorphism φ : Fn → FG maps C to a

left ideal (respectively a two-sided ideal) of FG. A left group code (respectively

a group code) is a linear code which is a left G-code (respectively a G-code)

for some group G. A (left) cyclic group code (respectively, abelian, metacyclic,

nilpotent group code, . . . ) is a linear code which is a (left) G-code for some

cyclic group (respectively, abelian, metacyclic, nilpotent group, . . . ) G. The

underlying group is not uniquely determined by the code itself. That means

that it is possible that a (left) non-abelian group code can also be realized as

an abelian group code.

61


We first state a result describing the (non-central) primitive idempotents of

finite semisimple group algebras of nilpotent groups. We opted to not include

the proof in this thesis because of its length and its technical behavior. The

proof can be found in [OVG11, Theorem 3.3].

Theorem 2.3.1 (Olteanu-Van Gelder)

Let F be a finite field and G a finite nilpotent group such that FG is

semisimple. Let (H,K) be a strong Shoda pair of G, C ∈ CF(H/K) and

set eC = eC(G,H,K), εC = εC(H,K), H/K = 〈a〉, E = EF(G,H/K).

Let E2/K and H2/K = 〈a2〉 (respectively E2′/K and H2′/K = 〈a2′〉)denote the 2-parts (respectively 2′-parts) of E/K and H/K respectively.

Then 〈a2′〉 has a cyclic complement 〈b2′〉 in E2′/K.

A complete set of orthogonal primitive idempotents of FGeC consists of

the conjugates of βeC = b2′β2εC by the elements of TeC = T2′T2TE , where

T2′ = {1, a2′ , a22′ , . . . , a

[E2′ :H2′ ]−12′ }, TE denotes a right transversal of E in

G and β2 and T2 are given according to the cases below.

1. If H2/K has a complement M2/K in E2/K then β2 = ”M2. More-

over, if M2/K is cyclic, then there exists b2 ∈ E2 such that E2/K is

given by the following presentation

〈a2, b2 : a22n = b2

2k = 1, a2b2 = a2

r〉,

and if M2/K is not cyclic, then there exist b2, c2 ∈ E2 such that

E2/K is given by the following presentation

〈a2, b2, c2 : a22n = b2

2k = c22 = 1, a2

b2 = a2r,

a2c2 = a2

−1, [b2, c2] = 1〉,

with r ≡ 1 mod 4 (or equivalently a22n−2

is central in E2/K).

a) T2 = {1, a2, a22, . . . , a

2k−12 }, if a2

2n−2

is central in E2/K (unless

n ≤ 1) and M2/K is cyclic; and

b) T2 = {1, a2, a22, . . . , a

d/2−12 , a2n−2

2 , a2n−2+12 , . . . , a

2n−2+d/2−12 },

where d = [E2 : H2], otherwise.

62


2. If H2/K does not have a complement in E2/K, then there exist

b2, c2 ∈ E2 such that E2/K is given by the following presentation

〈a2, b2, c2 : a22n = b2

2k = 1, c22 = a2

2n−1

,

a2b2 = a2

ra2c2 = a2

−1, [b2, c2] = 1〉,

with r ≡ 1 mod 4. In this case, β2 = “b2 1+xa2n−2

2 +ya2n−2

2 c22 and

T2 = {1, a2, a22, . . . , a

2k−12 , c2, c2a2, c2a

22, . . . , c2a

2k−12 },

with x, y ∈ F, satisfying x2 + y2 = −1 and y 6= 0.

Example 2.3.2 Consider once more the group G = C3 × Q8 from Exam-

ples 2.1.9 and 2.1.14, the field F5 and the strong Shoda pair (H,K), with

H = 〈a, x〉 and K = 1. Take C = {1, 5}. We computed before that E =

EF5(G,H/K) = H. Then E2 = H2 = 〈x〉 and E2′ = H2′ = 〈a〉. By The-

orem 2.3.1, F5GeC(G,H,K) has as a complete set of orthogonal primitive

idempotents the set

{εC(H,K), y−1εC(H,K)y}.

Theorem 2.3.1 provides a straightforward implementation in GAP. Neverthe-

less, in case 2, there might occur some difficulties finding solutions for the

equation x2 + y2 = −1 for x, y ∈ F and y 6= 0. However, it is possible to over-

come this problem. Note that here F has to be of odd order pn. If p ≡ 1 mod 4,

then y2 = −1 has a solution in Fp ⊆ F. Half of the elements α of Fp satisfy

the equation αp−12 = −1. So we can pick an α ∈ Fp at random and check if

the equality is satisfied. If not, repeat the process. When we have found such

an α, then take y = αp−14 and x = 0.

If 2 is a divisor of n, then y2 = −1 has a solution in Fp2 ⊆ F because

p2 ≡ 1 mod 4. Half of the elements β of Fp2 satisfy the equation βp2−1

2 = −1.

Pick a β ∈ Fp2 randomly. If the equality is satisfied, then take y = βp2−1

4 and

x = 0.

Now assume that p 6≡ 1 mod 4 and 2 - n. Recall that the Legendre symbol

(a/p) for an integer a and an odd prime number p, is defined as 1 if the con-

gruence x2 ≡ a mod p has a solution, as 0 if p divides a and as −1 otherwise.

Using the Legendre symbol, one can decide whether an element is a square

63


modulo p or not and this can be effectively calculated using the properties of

the Jacobi symbol as explained in standard references as, for example, in the

book of Kenneth Ireland and Michael Rosen [IR90].

Take now a random element a ∈ Fp ⊆ F and check if both a and −1 − aare squares in Fp. If so, then one can use the algorithm of Tonelli and Shanks

or the algorithm of Cornacchia to compute square roots modulo p and to

find x and y in Fp ⊆ F satisfying x2 + y2 = −1. For more information on

these algorithms the reader is referred to the literature, for example [Coh93,

Algorithms 1.5.1 and 1.5.2].

We illustrate this in an example.

Example 2.3.3 Consider the group with the following presentation

G =⟨a, b : a4 = 1, b12 = 1, b−1ab = a−1

⟩and its strong Shoda pair (H,K) with H =

⟨a, b2

⟩and K =

⟨a2b6

⟩. Then

H/K =⟨ab2K

⟩' C12 and I12(F7) ' Gal(F7(ζ12)/F7) = Gal(F49/F7) '

{1, 7}. Then EF7(G,H/K) = G since b−1ab2bK = (ab2)7K and 7 ∈ I12(F7).

Take C = {1, 7} ∈ CF7(G,H/K). We will compute a complete set of orthogonal

primitive idempotents in F7GeC(G,H,K).

Following Theorem 2.3.1, we compute E2 = G2 =⟨a, b3

⟩, E2′ = G2′ =⟨

b4⟩

= H2′ and H2 =⟨a, b6

⟩. Then H2/K does not have a complement in

G2/K and we are in case 2. We find the following presentation for G2/K:

〈ab6K, b3K : (ab6K)4 = 1, (b3K)2 = (ab6K)2,

(b3K)−1(ab6K)(b3K) = (ab6K)−1〉.

Since 2 ≡ 32 mod 7 and −3 ≡ 4 ≡ 52 mod 7, both 2 and −3 are squares in

F7 and hence we find that x = 3 and y = 5 satisfy the equation x2 + y2 = −1

in F7. We define T2 = {1, b3} and β2 = 1+3ab6+5ab9

2 . Now the set {β2, βb3

2 } is

a complete set of orthogonal primitive idempotents in F7GeC(G,H,K).

Computations involving strong Shoda pairs and primitive central idempo-

tents were already provided in the GAP package wedderga [BHK+14] and we

have included the function PrimitiveIdempotentsNilpotent implementing

the above algorithm.

Assume now that F is a finite field of order q and G is a finite group such

that the order of G is coprime to q. We give an analogue to Theorem 2.2.1

and focus on simple components of FG which are determined by a strong

Shoda pair (H,K) and a class C ∈ CF(H/K) such that τ(gH, g′H) = 1 for all

64


g, g′ ∈ E = EF(G,H/K). For such a component, we describe a complete set

of orthogonal primitive idempotents.

Before we do so, we need a normal element w ∈ F(ζ[H:K]) and a normal

basis {wx : x ∈ E/H} of

F(ζ[H:K])/F(ζ[H:K])E/H = Fqo/Fqo/[E:H]

(with o the multiplicative order of q modulo [H : K]). Recall that E/H, the

Galois group of Fqo over Fqo/[E:H] , is cyclic and generated by the Frobenius

automorphism x 7→ xqo/[E:H]

. Hence if w ∈ Fqo is such that the [E : H]

elements

{w,wqo/[E:H]

, . . . , w(qo/[E:H])[E:H]−1

}

are linearly independent, then this set forms a normal basis for Fqo over

Fqo/[E:H] . The existence of such a basis is stated in the Normal Basis The-

orem 1.3.1. For background on the construction of normal bases, see Emil

Artin [Art44], Heinz Luneburg [Lun86], Hendrik W. Jr. Lenstra [Len91] and

Shuhong Gao [Gao93]. The construction of normal bases for finite fields is also

implemented in GAP in the method NormalBase.

The proof of the following theorem is very similar to the one of Theorem 2.2.1

and is therefore omitted.

Theorem 2.3.4

Let G be a finite group and F a finite field of order q such that q is coprimeto the order of G. Let (H,K) be a strong Shoda pair of G such thatτ(gH, g′H) = 1 for all g, g′ ∈ E = EF(G,H/K), and let C ∈ CF(H/K).Let ε = εC(H,K) and e = eC(G,H,K). Let w be a normal element ofFqo/Fqo/[E:H] (with o the multiplicative order of q modulo [H : K]) andB the normal basis determined by w. Let ψ be the isomorphism betweenFEε and the matrix algebra M[E:H](Fqo/[E:H]) with respect to the basisB as stated in Theorem 1.5.2. Let P,A ∈M[E:H](Fqo/[E:H]) be defined asfollows:

P =

1 1 1 · · · 1 1

1 −1 0 · · · 0 0

1 0 −1 · · · 0 0...

......

. . ....

...

1 0 0 · · · −1 0

1 0 0 · · · 0 −1

and A =

0 0 · · · 0 1

1 0 · · · 0 0

0 1 · · · 0 0...

.... . .

......

0 0 · · · 0 0

0 0 · · · 1 0

.

65


Then

{xT1εx−1 : x ∈ T2 〈xe〉}

is a complete set of orthogonal primitive idempotents of FGe where xe =

ψ−1(PAP−1), T1 is a transversal of H in E and T2 is a right transversal

of E in G. By T1 we denote the element 1|T1|

∑t∈T1

t in FG.

We have included an implementation of the above theorem in the function

PrimitiveIdempotentsTrivialTwisting in wedderga. This was possible be-

cause GAP can easily find a normal basis and we can compute ψ−1(PAP−1)

algorithmically.

Using our implementation in wedderga of primitive idempotents of finite

semisimple group algebras described in Theorems 2.3.1 and 2.3.4, it is possible

to construct many left group codes. For more details and examples on this

topic, the reader is referred to our joined work with Gabriela Olteanu and the

references given in [OVG15].

2.4 conclusions

We made some progress on the description of non-central primitive idempo-

tents of rational group algebras QG and of finite semisimple group algebras

FG. However, there are still cases to study, for example to cover all classes of

non-nilpotent groups within the strongly monomial groups or to replace the

field Q or F with a number field F .

Also for the description of primitive central idempotents one is somehow

limited to strongly monomial groups. The ultimate goal would be to study

idempotents in rational group algebras of all finite groups, including simple

groups. Since all our constructions are based on the idempotent “N , for a

non-trivial normal subgroup N , none of our methods apply to simple groups.

However, some recent results (see [JOdR12, Jan13, Olt07]) show that the com-

putation of the primitive central idempotents of QG for G a finite arbitrary

group can be reduced to the case of finite strongly monomial groups, which

means that this case is essential for the computation of both the primitive

central idempotents and the Wedderburn decomposition of a rational group

algebra.

66

3E XC E P T I O N A L C O M P O N E N T S

When studying the group of units of RG for a finite group G and the ring of

integers R of a number field F , one is often restricted to groups such that no

exceptional component appears in the Wedderburn decomposition of FG. A

good example for this is Proposition 1.13.3.

Therefore, it is useful to classify finite groups G and abelian number fields F

such that FG contains an exceptional component in its Wedderburn decompo-

sition. Recall from Corollary 1.9.9, that we know exactly which isomorphism

types of exceptional components can occur.

For the unit groups of exceptional components of type EC1 very little is

known. Let O be a Z-order in a non-commutative division ring different from

a totally definite quaternion algebra. Then SL1(O) is infinite and to the best of

our knowledge, there are no generic constructions of subgroups of finite index

known. In 2000, Kleinert [Kle00b] gave a commendable survey on that topic.

Up to that date no constructions were known for degree 3 division rings and

also for degree 2 very little was known. Only recently, there was some progress

made by [CJLdR04, JJK+15] for degree 2 division rings. Braun, Coulangeon,

Nebe and Schonnenbeck provided a generalization of Voronoı’s algorithm to

tackle the problem [BCNS], they give examples to work with division algebras

of degree 2 and 3. Still one would like to have generic constructions, and more-

over to have constructions of groups of units in RG that contain a subgroup

of finite index in U(O), when O is a Z-order in a division ring appearing in

the Wedderburn decomposition of FG.

In this chapter we first classify all exceptional components of type EC2

occurring in the Wedderburn decomposition of group algebras of finite groups

over arbitrary number fields. We do this by giving a full list of finite groups

G, number fields F and exceptional components M2(D) such that M2(D) is a

faithful Wedderburn component of FG, cf. Theorem 3.1.2.

67

exceptional components

Afterwards we deal with exceptional components of type EC1. We classify

F -critical groups, i.e. groups G such that FG has an exceptional component

of type EC1 in its Wedderburn decomposition, but no proper quotient has this

property. In this way we obtain a minimal list of exceptional components of

type EC1 appearing in group algebras FG for abelian number fields F and

G finite. Regarding the scale of the difficulty of the problem, at least for the

division rings in the list the unit groups of Z-orders have to be studied. Note

that any group H such that FH has a non-commutative division ring (not

totally definite quaternion) in its Wedderburn decomposition has an epimor-

phic F -critical image G such that if an exceptional component D of type EC1

appears as a faithful Wedderburn component of FG, then also FH has D

as a simple component. Having an F -critical epimorphic image for a group

implies that, up to now, there is no hope for a generic construction of units

in RG. We give necessary and sufficient conditions for a finite group G to be

F -critical expressed in easy arithmetic formulas in terms of parameters of the

group, and ramification indices and residue degrees of extensions of F , rely-

ing on the parameters of the group. For any abelian number field F and any

finite F -critical group G, we explicitly describe the division ring, which is an

exceptional component of type EC1 in FG, cf. Theorem 3.2.21.

All results presented in this chapter extend a result of Mauricio Caicedo

and Angel del Rıo [CdR14] where they handled the case F = Q. The results

of Theorems 1.9.6, 1.9.8 and 1.10.3 and Proposition 1.9.7 give more insight in

the exceptional components of type EC2, therefore it is natural to distinguish

between exceptional components of type EC1 and type EC2, in contrast to

what was done in [CdR14]. So our work yields (also for the case of rational

group algebras) an extension of the above cited result.

3.1 group algebras with exceptional components of type

ec2

We call a Wedderburn component A of FG faithful if G is faithfully embedded

in A via the Wedderburn isomorphism.

In this section, we give a full list of finite groups G and number fields

F having faithful exceptional components of type EC2 in FG. Employing

Lemma 3.1.1 one can deduce from this list all exceptional components of type

EC2 within group algebras over number fields.

68

3.1 group algebras with exceptional components of type ec2

Lemma 3.1.1

Let G be a finite group, F be a number field and ρ an irreducible F -

representation of G. Let e be the primitive central idempotent associated

to ρ and K be the kernel of ρ. Then the group Ge is faithfully embedded

in (FG)e and its F -span equals (FG)e. In particular, Ge is isomorphic

to a subgroup of U((FG)e).

Proof Consider the irreducible representation ρ : G → U(FGe). This in-

duces a faithful representation ρ : G/K → U(FGe) and G/K ' ρ(G) = Ge.

Since FGe is the F -span of ρ(G) = ρ(G/K) and FGe is simple, FGe is also

isomorphic to a Wedderburn component of F (G/K).

Theorem 3.1.2

Let F be a number field, G be a finite group and B a simple exceptional

algebra of type EC2. Then B is a faithful Wedderburn component of FG

if and only if G, F , B is a row listed in Table 2.

Proof Let B be a simple exceptional algebra of type EC2 and assume that

B is a faithful Wedderburn component of FG, then by Lemma 3.1.1, G is a

subgroup of U(B) and B is isomorphic to an algebra stated in Theorems 1.9.6

and 1.9.8.

The subgroups of M2(Q),M2(Q(√−1)),M2(Q(

√−2)),M2(Q(

√−3)) are em-

bedded in GL(2, 25) by Proposition 1.9.7. The maximal finite subgroups of

2×2-matrices over totally definite quaternion algebras with center Q were clas-

sified in Theorem 1.10.3. They can be accessed in Magma [BCP97] by calling

QuaternionicMatrixGroupDatabase().

It is also clear that when FG has a Wedderburn component B then F is

contained in the center of B, which restricts the possibilities of F for G and

B fixed.

Additionally, using the GAP-package wedderga, one can compute a finite list

of groups G that have B as a faithful component over F . We mainly use the

function WedderburnDecompositionWithDivAlgParts which returns the size

of the matrices, the centers and the local indices of all Wedderburn components

of a group algebra and allows us to compare the Wedderburn components to

the possibilities of B above. This is possible, since F is a number field and the

isomorphism type of division algebras is determined by its list of local Hasse

invariants at all primes of F (Theorem 1.4.2). For quaternion algebras the

local Hasse invariants are uniquely determined by the local Schur indices.

69


Notation 3.1.3 (in Table 2) We use the GAP notation for the group struc-

ture and the identification number from the SmallGroups library. For a non-

split extension of A by B, we write A.B. If an exceptional component appears

several times in FG, this multiplicity is indicated in the last column.

Table 2: List of all groups having a faithful exceptional component of type EC2

ID Structure F B

[6, 1] S3 Q 1× M2 (Q)

[6, 1] S3 Q(√−1) 1× M2

(Q(√−1))

[6, 1] S3 Q(√−2) 1× M2

(Q(√−2))

[6, 1] S3 Q(√−3) 1× M2

(Q(√−3))

[8, 3] D8 Q 1× M2 (Q)

[8, 3] D8 Q(√−1) 1× M2

(Q(√−1))

[8, 3] D8 Q(√−2) 1× M2

(Q(√−2))

[8, 3] D8 Q(√−3) 1× M2

(Q(√−3))

[8, 4] Q8 Q(√−1) 1× M2

(Q(√−1))

[8, 4] Q8 Q(√−2) 1× M2

(Q(√−2))

[8, 4] Q8 Q(√−3) 1× M2

(Q(√−3))

[12, 1] C3 o C4 Q(√−1) 1× M2

(Q(√−1))

[12, 1] C3 o C4 Q(√−3) 1× M2

(Q(√−3))

[12, 4] D12 Q 1× M2 (Q)

[12, 4] D12 Q(√−1) 1× M2

(Q(√−1))

[12, 4] D12 Q(√−2) 1× M2

(Q(√−2))

[12, 4] D12 Q(√−3) 1× M2

(Q(√−3))

[16, 6] C8 o C2 Q 1× M2

(Q(√−1))

[16, 6] C8 o C2 Q(√−1) 2× M2

(Q(√−1))

[16, 8] QD16 (also denoted by D−16) Q 1× M2

(Q(√−2))

[16, 8] QD16 (also denoted by D−16) Q(

√−2) 2× M2

(Q(√−2))

[16, 13] (C4 × C2)o C2 Q 1× M2

(Q(√−1))

[16, 13] (C4 × C2)o C2 Q(√−1) 2× M2

(Q(√−1))

[18, 3] C3 × S3 Q 1× M2

(Q(√−3))

[18, 3] C3 × S3 Q(√−3) 2× M2

(Q(√−3))

[24, 1] C3 o C8 Q 1× M2

(Q(√−1))

[24, 1] C3 o C8 Q(√−1) 2× M2

(Q(√−1))

[24, 3] SL(2, 3) Q 1× M2

(Q(√−3))

[24, 3] SL(2, 3) Q(√−1) 1× M2

(Q(√−1))

[24, 3] SL(2, 3) Q(√−2) 1× M2

(Q(√−2))

continued

70


ID Structure F B

[24, 3] SL(2, 3) Q(√−3) 3× M2

(Q(√−3))

[24, 5] C4 × S3 Q 1× M2

(Q(√−1))

[24, 5] C4 × S3 Q(√−1) 2× M2

(Q(√−1))

[24, 8] (C6 × C2)o C2 Q 1× M2

(Q(√−3))

[24, 8] (C6 × C2)o C2 Q(√−3) 2× M2

(Q(√−3))

[24, 10] C3 ×D8 Q 1× M2

(Q(√−3))

[24, 10] C3 ×D8 Q(√−3) 2× M2

(Q(√−3))

[24, 11] C3 ×Q8 Q 1× M2

(Q(√−3))

[24, 11] C3 ×Q8 Q(√−3) 2× M2

(Q(√−3))

[32, 8] (C2 × C2).(C4 × C2) Q 1× M2

(−1,−1Q

)[32, 11] (C4 × C4)o C2 Q 2× M2

(Q(√−1))

[32, 11] (C4 × C4)o C2 Q(√−1) 4× M2

(Q(√−1))

[32, 44] (C2 ×Q8)o C2 Q 1× M2

(−1,−1Q

)[32, 50] (C2 ×Q8)o C2 Q 1× M2

(−1,−1Q

)[36, 6] C3 × (C3 o C4) Q 1× M2

(Q(√−3))

[36, 6] C3 × (C3 o C4) Q(√−3) 2× M2

(Q(√−3))

[36, 12] C6 × S3 Q 1× M2

(Q(√−3))

[36, 12] C6 × S3 Q(√−3) 2× M2

(Q(√−3))

[40, 3] C5 o C8 Q 1× M2

(−2,−5Q

)[48, 16] (C3 o C8)o C2 Q 1× M2

(−1,−1Q

)[48, 18] C3 oQ16 Q 1× M2

(−1,−3Q

)[48, 28] SL(2, 3).C2 = O∗ Q 1× M2

(−1,−3Q

)[48, 29] GL(2, 3) Q 1× M2

(Q(√−2))

[48, 29] GL(2, 3) Q(√−2) 2× M2

(Q(√−2))

[48, 33] SL(2, 3)o C2 Q 1× M2

(Q(√−1))

[48, 33] SL(2, 3)o C2 Q(√−1) 2× M2

(Q(√−1))

[48, 39] (C2 × (C3 o C4))o C2 Q 1× M2

(−1,−3Q

)[48, 40] Q8 × S3 Q 1× M2

(−1,−1Q

)[64, 37] (C4 × C2).(C4 × C2) Q 2× M2

(−1,−1Q

)[64, 137] ((C4 × C4)o C2)o C2 Q 2× M2

(−1,−1Q

)[72, 19] (C3 × C3)o C8 Q 2× M2

(−1,−3Q

)[72, 20] (C3 o C4)× S3 Q 1× M2

(−1,−3Q

)[72, 22] (C6 × S3)o C2 Q 1× M2

(−1,−3Q

)[72, 24] (C3 × C3)oQ8 Q 1× M2

(−1,−3Q

)[72, 25] C3 × SL(2, 3) Q 3× M2

(Q(√−3))

[72, 25] C3 × SL(2, 3) Q(√−3) 6× M2

(Q(√−3))

[72, 30] C3 × ((C6 × C2)o C2) Q 2× M2

(Q(√−3))

continued

71


ID Structure F B

[72, 30] C3 × ((C6 × C2)o C2) Q(√−3) 4× M2

(Q(√−3))

[96, 67] SL(2, 3)o C4 Q 2× M2

(Q(√−1))

[96, 67] SL(2, 3)o C4 Q(√−1) 4× M2

(Q(√−1))

[96, 190] (C2 × SL(2, 3))o C2 Q 1× M2

(−1,−1Q

)[96, 191] (SL(2, 3).C2 = O∗)o C2 Q 1× M2

(−1,−1Q

)[96, 202] (C2 × SL(2, 3))o C2 Q 1× M2

(−1,−1Q

)[120, 5] SL(2, 5) Q 1× M2

(−1,−3Q

)[128, 937] (Q8 ×Q8)o C2 Q 4× M2

(−1,−1Q

)[144, 124] C3 o (SL(2, 3).C2 = O∗) Q 3× M2

(−1,−3Q

)[144, 128] S3 × SL(2, 3) Q 1× M2

(−1,−1Q

)[144, 135] ((C3 × C3)o C8)o C2 Q 4× M2

(−1,−3Q

)[144, 148] (C2 × ((C3 × C3)o C4))o C2 Q 2× M2

(−1,−3Q

)[160, 199] ((C2 ×Q8)o C2)o C5 Q 1× M2

(−1,−1Q

)[192, 989] ((SL(2, 3).C2 = O∗)o C2)o C2 Q 2× M2

(−1,−1Q

)[240, 89] C2.S5 = SL(2, 5).C2 Q 1× M2

(−2,−5Q

)[240, 90] SL(2, 5)o C2 Q 1× M2

(−2,−5Q

)[288, 389] ((C3 o C4)× (C3 o C4))o C2 Q 2× M2

(−1,−3Q

)[320, 1581] (((C2 ×Q8)o C2)o C5).C2 Q 2× M2

(−1,−1Q

)[384, 618] ((Q8 ×Q8)o C3)o C2 Q 1× M2

(−1,−1Q

)[384, 18130] ((Q8 ×Q8)o C3)o C2 Q 1× M2

(−1,−1Q

)[720, 409] SL(2, 9) Q 2× M2

(−1,−3Q

)[1152, 155468] (SL(2, 3)× SL(2, 3))o C2 Q 1× M2

(−1,−1Q

)[1920, 241003] C2.((C2 × C2 × C2 × C2)oA5) Q 1× M2

(−1,−1Q

)3.2 group algebras with exceptional components of type

ec1

In this section we consider exceptional components of type EC1. We provide

necessary and sufficient conditions for a finite group G to be F -critical.

Let G be a finite group and F an abelian number field. We say that G is

F -critical if and only if

1. FG has a Wedderburn component which is exceptional of type EC1, and

2. for any 1 6= N �G the group algebra F (G/N) does not have a Wedder-

burn component which is exceptional of type EC1.

72


Note that if a group G is F -critical with corresponding exceptional compo-

nent B, the F -representation of G associated to B is necessarily faithful. In

particular G is in the classification of Amitsur (cf. Theorem 1.10.1).

Lemma 3.2.1

A division algebra A is exceptional if and only if ind(A) > 2 or ind(A) = 2

and m∞(A) 6= 2.

Proof This follows immediately since a quaternion algebra is totally definite

if and only if its local index at infinity is 2.

We recall that for a finite Galois extension F/Q, always F is either totally

real or totally imaginary.

Proposition 3.2.2

Let A =Äa,bK

äwith K a totally real finite Galois extension of Q, a, b ∈ K

totally negative and let F be a finite Galois extension of Q containing K.

Then F ⊗K A is a division algebra if and only if F is a totally real number

field or there exists a prime number p such that mp(A) 6= 1 and both

ep(F/K) and fp(F/K) are odd.

Proof Let B = F ⊗K A. Note that B is a division algebra if and only if

B is not split, as the degree is 2. Furthermore, B is not split if and only if

mp(B) 6= 1 for some prime p of Q (finite or infinite).

Assume that F is totally real, then there exists a real embedding σ of F

and for the completion of this embedding we find R⊗F B 'Äσ(a),σ(b)

R

ä, hence

m∞(B) = 2 and B is a division algebra.

Assume that F is totally imaginary, then for all embeddings of F all local

Schur indices for infinite primes are 1, since the completion of any embedding

of F is C and splits B.

Fix a finite prime number p. If mp(A) = 1, then clearly mp(B) = 1. If

mp(A) 6= 1, then it is 2 and due to Theorem 1.4.8, mp(B) = 1 if and only if

mp(A) | [Fp : ›Kp]. Hence the result follows since [Fp : ›Kp] = ep(F/K)fp(F/K)

by Proposition 1.4.6.

Proposition 3.2.2 is a generalization of the following statement from [SW86,

Theorem 2.1.9].

73


Corollary 3.2.3

For F a finite Galois extension of Q, F ⊗QÄ−1,−1

Q

äis a division algebra

if and only if F is a totally real field or both e2(F/Q) and f2(F/Q) are

odd.

Proof Let A =Ä−1,−1

Q

ä. Then m2(A) = 2 and mp(A) = 1 for all odd

prime numbers p, see Example 1.4.12. The result easily follows from Proposi-

tion 3.2.2.

We first consider the NZ-groups from Amitsur’s classification in Proposi-

tions 3.2.4 to 3.2.7 and 3.2.9.

Proposition 3.2.4

Let F be an abelian number field and let O∗ be the binary octahedral

group. Then O∗ is never F -critical.

Proof The Wedderburn decomposition of QO∗ equals

2Q⊕M2(Q)⊕ 2M3(Q)⊕Ç−1,−1

Q(√

2)

å⊕M2

Å−1,−3

Q

ã.

Hence the only possible exceptional component of type EC1 of FO∗ can come

from

F ⊗Q

Ç−1,−1

Q(√

2)

å=

Ç−1,−1

F (√

2)

å[F∩Q(√

2):Q]

.

The quaternion algebra(−1,−1

Q(√

2)

)has all local Schur indices 1, except the local

index m∞

(−1,−1

Q(√

2)

)= 2. By Proposition 3.2.2,

(−1,−1

F (√

2)

)is therefore only a

division algebra when F is totally real. But in this case it is a totally definite

quaternion algebra and hence not exceptional.

Proposition 3.2.5

Let F be an abelian number field. Then SL(2, 5) is never F -critical.

Proof The Wedderburn decomposition of QSL(2, 5) equals

Q⊕M4(Q)⊕Ç−1,−1

Q(√

5)

å⊕M2

Å−1,−3

Q

ã⊕M5(Q)⊕M3

Å−1,−1

Q

ã⊕M3(Q(

√5)).

74


The only possible exceptional component of type EC1 of FSL(2, 5) can come

from

F ⊗Q

Ç−1,−1

Q(√

5)

å=

Ç−1,−1

F (√

5)

å[F∩Q(√

5):Q]

.

The algebra(−1,−1

Q(√

5)

)has all local Schur indices 1, except m∞

(−1,−1

Q(√

5)

)= 2.

By Proposition 3.2.2,(−1,−1

F (√

5)

)is therefore only a division algebra when F is

totally real. But in this case it is a totally definite quaternion algebra and

hence not exceptional.

Proposition 3.2.6

Let F be an abelian number field and Q4k be the generalized quaternion

group with k even. Then Q4k is F -critical if and only if k = 2, F is totally

imaginary and both e2(F/Q) and f2(F/Q) are odd. In this case FQ8

contains an exceptional component(−1,−1

F

).

Proof Let Q4k =⟨a, b : a2k = 1, ak = b2, b−1ab = a−1

⟩with k = 2tk′, t ≥ 1

and 2 - k′, then by Lemma 1.7.17, the non-commutative components of QQ4k

come from the strong Shoda pairs (〈a〉 ,⟨ad⟩) with d | 2k such that d 6= 1, 2.

The corresponding simple components are

QQ4ke(Q4k, 〈a〉 ,⟨ad⟩)

=

{(Q(ζd)/Q(ζd + ζ−1

d ), 1) = M2(Q(ζd + ζ−1d )), if d | k

(Q(ζd)/Q(ζd + ζ−1d ),−1) =

(−1,−1

Q(ζd+ζ−1d

)

), otherwise.

(10)

We claim that k′ = 1. For this, suppose that Q4k is F -critical and k′ 6= 1.

By the above paragraph the exceptional component of FQ4k comes from

F ⊗Q

Ç−1,−1

Q(ζd + ζ−1d )

å=

Ç−1,−1

F (ζd + ζ−1d )

å[F∩Q(ζd+ζ−1d

):Q]

=

ÅF (ζd + ζ−1

d )⊗Q

Å−1,−1

Q

ãã[F∩Q(ζd+ζ−1d

):Q]

for some d satisfying that d | 2k and d - k. Note that this implies that

d = 2t+1k′1 where k′1 | k′ and 2 - k′1. This quaternion algebra cannot be

totally definite (i.e. F is not totally real) although it is a division algebra.

Hence, by Corollary 3.2.3, e2(F (ζd + ζ−1d )/Q) and f2(F (ζd + ζ−1

d )/Q) are

75


odd. Due to k′ 6= 1, we have that Q4·2t is a non-abelian proper quotient of

Q4k, moreover

Å−1,−1

F (ζ2t+1+ζ−1

2t+1)

ãis a simple component of FQ4·2t . Hence it

must be either a totally definite quaternion algebra or a 2 × 2-matrix ring

over F (ζ2t+1 + ζ−12t+1). If F splits the quaternion algebra

Å−1,−1

Q(ζ2t+1+ζ−1

2t+1)

ã, by

Corollary 3.2.3, e2(F (ζ2t+1 +ζ−12t+1)/Q) or f2(F (ζ2t+1 +ζ−1

2t+1)/Q) is even. Since,

e2(F (ζd + ζ−1d )/Q)

= e2(F (ζd + ζ−1d )/F (ζ2t+1 + ζ−1

2t+1))e2(F (ζ2t+1 + ζ−12t+1)/Q)

and

f2(F (ζd + ζ−1d )/Q)

= f2(F (ζd + ζ−1d )/F (ζ2t+1 + ζ−1

2t+1))f2(F (ζ2t+1 + ζ−12t+1)/Q),

this implies that e2(F (ζd + ζ−1d )/Q) or f2(F (ζd + ζ−1

d )/Q) is even, a contra-

diction. Therefore

Å−1,−1

F (ζ2t+1+ζ−1

2t+1)

ãis a totally definite quaternion algebra,

and then F (ζ2t+1 + ζ−12t+1) is totally real, and hence F is totally real, again a

contradiction. Therefore k′ = 1 and, by formula (10),

Å−1,−1

F (ζ2t+1+ζ−1

2t+1)

ãis the

only candidate for being an exceptional component of type EC1.

We claim that t = 1. By Corollary 3.2.3,

Å−1,−1

F (ζ2t+1+ζ−1

2t+1)

ãis a division

algebra whenever e2(F (ζ2t+1 +ζ−12t+1)/Q) and f2(F (ζ2t+1 +ζ−1

2t+1)/Q) are odd or

F is totally real. Assume that t > 1, then F (ζ8 + ζ−18 ) ⊆ F (ζ2t+1 + ζ−1

2t+1) and

e2(F (ζ8 + ζ−18 )/Q) = e2(Q(

√2)/Q) = 2 and hence also e2(F (ζ2t+1 + ζ−1

2t+1)/Q)

is even. But in this case

Å−1,−1

F (ζ2t+1+ζ−1

2t+1)

ãis a totally definite quaternion

algebra and hence not exceptional. We conclude that t must be equal to 1

and k = 2. Since G is F -critical,(−1,−1

F

)is not a totally definite quaternion

algebra. Hence F is totally imaginary and e2(F/Q) and f2(F/Q) are odd.

Finally we prove the converse. By the assumptions and Corollary 3.2.3, we

have that(−1,−1

F

)is an exceptional component of type EC1 of FQ8. The only

proper quotients of Q8 are abelian groups and hence Q8 is F -critical.

As we have seen, the reciprocity rules from Theorem 1.4.4 are very useful

and often lead to an answer in a few steps. We also get an advantage of the

use of those rules in the remaining cases (NZ)(d) and (NZ)(e).

76


Proposition 3.2.7

Let F be an abelian number field.

1. SL(2, 3) is F -critical if and only if F is totally imaginary and both

e2(F/Q) and f2(F/Q) are odd. In this case, FSL(2, 3) contains an

exceptional component(−1,−1

F

).

2. Let M be a group in (Z) of order coprime to 6, such that 2 has

odd order modulo |M |. Then SL(2, 3)×M is F -critical if and only

if M is a cyclic group of prime order p, F is totally real and both

e2(F (ζp)/Q) and f2(F (ζp)/Q) are odd. In this case, F (SL(2, 3)×Cp)contains an exceptional component

Ä−1,−1F (ζp)

ä.

Proof 1. Let G = SL(2, 3). We first assume that G is F -critical. The

Wedderburn decomposition of QG equals

QG ' Q⊕Q(ζ3)⊕M3(Q)⊕Å−1,−1

Q

ã⊕M2(Q(ζ3)).

The only exceptional component of type EC1 of FG can be F ⊗QÄ−1,−1

Q

ä=(−1,−1

F

), hence it is an exceptional component of FG. Using Corollary 3.2.3

the result follows.

Now we prove the converse. By Corollary 3.2.3 we have that(−1,−1

F

)is an

exceptional component of FG of type EC1. On the other hand, G has only one

non-abelian proper quotient which is isomorphic to A4, and the Wedderburn

decomposition of QA4 equals

QA4 ' Q⊕Q(ζ3)⊕M3(Q).

Hence the group algebra of any non-abelian proper quotient of G does not

have exceptional components. So we conclude that G is F -critical.

2. Let G = SL(2, 3)×M . We first assume that G is F -critical. Observe that

in the Wedderburn decomposition of FM totally definite quaternion algebras

can not appear, since the order of M is odd. Moreover, due to the fact that

M is a proper quotient of G, FM does not have non-commutative division

algebras as simple components. Another non-abelian quotient of G is SL(2, 3),

and according to the Wedderburn decomposition of QSL(2, 3),(−1,−1

F

)has to

be either a totally definite quaternion algebra or a 2 × 2-matrix ring over

F . Suppose that(−1,−1

F

)is a 2 × 2-matrix ring over F . The fact that

77


FG ' FSL(2, 3) ⊗F FM implies that there is not any division algebra in

the Wedderburn decomposition of FG which is a contradiction. Therefore,(−1,−1F

)is a totally definite quaternion algebra and F is a totally real field.

On the other hand, let D be an exceptional component of type EC1 of FG,

then D ' D1 ⊗F D2 where D1 and D2 are simple components of FSL(2, 3)

and FM respectively. Having in mind the Wedderburn decompositions of

FSL(2, 3) and FM , and since D is a division algebra which is not a totally

definite quaternion algebra, we can deduce that D1 '(−1,−1

F

)and D2 ' F (ζd)

for some divisor d of the order of M , d 6= 1. We know that F (ζd) is a simple

component of F (M/M ′) (and so D is a simple component of F (G/M ′)), hence

by hypothesis M ′ is trivial, so that M is abelian and by the conditions in

Theorem 1.10.1, M is a cyclic group. Now we claim that M has prime order.

Let d be a proper divisor of |M |, then(−1,−1

F

)⊗F F (ζd) is a simple component

of F (G/C|M |/d). By hypothesis it must be a 2 × 2-matrix ring over F (ζd),

and it follows that D '(−1,−1

F

)⊗F F (ζ|M |). By Corollary 3.2.3, F (ζd) is

a totally imaginary field and e2(F (ζd)/Q) or f2(F (ζd)/Q) is even, moreover

both e2(F (ζ|M |)/Q) and f2(F (ζ|M |)/Q) are odd. By Theorem 1.4.4, we have

e2(F (ζ|M |)/Q) = e2(F (ζ|M |)/F (ζd))e2(F (ζd)/Q)

and

f2(F (ζ|M |)/Q) = f2(F (ζ|M |)/F (ζd))f2(F (ζd)/Q)

are both odd, but this is a contradiction since e2(F (ζd)/Q) or f2(F (ζd)/Q) is

even. So the claim follows.

By the above paragraph, we have that G = SL(2, 3)× Cp and

D 'Å−1,−1

F

ã⊗F F (ζp)

is an exceptional component of type EC1 of FG. Then again by Corollary 3.2.3

both e2(F (ζp)/Q) and f2(F (ζp)/Q) are odd.

Now suppose that G = SL(2, 3) × Cp. Using Corollary 3.2.3 and having in

mind the Wedderburn decomposition of QSL(2, 3), we can deduce that D '(−1,−1F

)⊗F F (ζp) is the unique exceptional component of FG of type EC1.

Note that the non-abelian proper quotients of G are SL(2, 3), A4 and A4×Cp.Due to the fact that F is a totally real field, FSL(2, 3) does not have exceptional

components of type EC1. As in the Wedderburn decomposition of QA4 and

Q(A4 × Cp) only fields and matrix rings show up, we have that FA4 and

F (A4×Cp) do not have division algebras as simple components. This finishes

the proof.

78


Remark 3.2.8 Assume that p is an odd prime number with o2(p) odd. By

Theorem 1.4.9, e2(Q(ζp)/Q) = 1 and f2(Q(ζp)/Q) = op(2) is odd. By Theo-

rem 1.4.4, it follows that e2(F (ζp)/Q) and f2(F (ζp)/Q) being odd is equiva-

lent with both the ramification index e2(F (ζp)/Q(ζp)) and the residue degree

f2(F (ζp)/Q(ζp)) being odd.

Following the proof of the previous proposition we have:

Proposition 3.2.9

Let M be a group in (Z) of odd order such that 2 has odd order modulo

|M | and let F be an abelian number field. Then Q8 ×M is F -critical if

and only if M is a cyclic group of prime order p, F is totally real and both

e2(F (ζp)/Q) and f2(F (ζp)/Q) are odd. In this case F (Q8 ×Cp) contains

an exceptional componentÄ−1,−1F (ζp)

ä.

Now we consider the Z-groups from Amitsur’s classification. Amitsur sug-

gests in his paper [Ami55] that one can discover a minimal faithful division

algebra component in the Wedderburn decomposition of the groups in (Z).

We give some lemmas first. Recall that we denote by Im(F ) the image of

Gal(F (ζm)/F ) in U(Z/mZ). Because of this correspondence, we will often

abuse notation and denote an automorphism Q(ζm) → Q(ζm) : ζm 7→ ζrm by

its defining integer r.

Let G = 〈a〉m ok 〈b〉n and A =⟨abn/k

⟩. Then A is cyclic and normal

and maximal abelian in G and hence (A, 1) is a strong Shoda pair of G. We

investigate the structure of the simple algebra FGeC(G,⟨abn/k

⟩, 1), described

in Theorem 2.1.6, when EF (G,⟨abn/k

⟩) = G.

Lemma 3.2.10

Let F be a number field and G = 〈a〉m ok 〈b〉n with gcd(m,n) = 1,

b−1ab = ar and r an integer with om(r) = nk . Then EF (G,

⟨abn/k

⟩) = G

if and only if Q(ζm) ∩ F is contained in Q(ζm)〈r〉.

Proof Let r′ be such that r′ ≡ r mod m and r′ ≡ 1 mod k. Then, G =

EF (G,⟨abn/k

⟩) if and only if 〈r′〉 ⊆ Imk(F ). This happens if and only if

σ : ζmk 7→ ζr′

mk is a Galois automorphism of the extension F (ζmk)/F , which is

equivalent with F being in the fixed field of σ. Since σ fixes ζk, this again is

equivalent with Q(ζm) ∩ F ⊆ Q(ζm)〈r〉.

79


Lemma 3.2.11

Let F be a number field and G = 〈a〉m ok 〈b〉n with gcd(m,n) = 1,

b−1ab = ar, r an integer such that om(r) = nk and Q(ζm)∩F ⊆ Q(ζm)〈r〉.

Let A =⟨abn/k

⟩, C ∈ CF (A), σ : F (ζmk) → F (ζmk) : ζm 7→ ζrm and

K = F (ζmk)〈σ〉. Then

FGeC(G,A, 1) = (K(ζm)/K, σ, ζk).

If furthermore n = 4 and k = 2, then r ≡ −1 mod m and

FGeC(G,A, 1) =

Å−1, (ζm − ζ−1m )2

F (ζm + ζ−1m )

ã.

Proof Let A =⟨abn/k

⟩. From Lemma 3.2.10 and Theorem 2.1.6 it follows

that

FGeC(G,A, 1) = F (ζmk) ∗G/A =

nk−1∑i=0

F (ζmk)uσi

with uσiζm = ζri

muσi , uσiζk = ζkuσi and un/kσi = ζk. The center of this simple

algebra is clearly equal to K = F (ζmk)〈σ〉 and therefore we can denote it as

the cyclic cyclotomic algebra FGeC(G,A, 1) = (K(ζm)/K, σ, ζk).

If n = 4 and k = 2, then the degree of FGeC(G,A, 1) is 2 and hence it is a

quaternion algebra over its center F (ζm + ζ−1m ). An easy computation shows

that FGeC(G,A, 1) is generated over its center by elements x and y satisfying

x2 = ζ2 = −1 and y2 = (ζm − ζ−1m )2 and xy = −yx.

The following lemma comes from [CdR14, Lemma 2.3].

Lemma 3.2.12 (Caicedo-del Rıo)

Let G = 〈a〉m ok 〈b〉n with gcd(m,n) = 1, b−1ab = ar, r an integer such

that om(r) = nk and let A =

⟨abn/k

⟩. Then G is a subgroup of a division

algebra D if and only if QGe(G,A, 1) is embedded in D.


⟩. By Lemma 3.2.11,

QGe(G,A, 1) = Q(ζmk) ∗G/A =

nk−1∑i=0

Q(ζmk)uσi ,

80


with un/kσi = ζk, uσiζm = ζr

i

muσi and uσiζk = ζkuσi . Moreover, a 7→ ζm and

b 7→ uσ determines an injective group homomorphism

G→ U(QGe(G,A, 1)).

Let D be a division algebra such that f : G → U(D) is an injective group

homomorphism. Then f(a) and f(bn/k) are commuting roots of unity of order

m and k respectively and f(b−1ab) = f(a)r. Thus ζm 7→ f(a) and uσ 7→ f(b)

determines an algebra homomorphism Q(ζmk) ∗ G/A → D which is injective

because Q(ζmk) ∗G/A is simple.

Proposition 3.2.13

Let F be a number field and G = 〈a〉m ok 〈b〉n with gcd(n,m) = 1. Let

A =⟨abn/k

⟩and C ∈ CF (A). If FGeC(G,A, 1) is not a division algebra,

then FG does not have a division algebra as a faithful simple component.

Furthermore, if FG contains a division algebra D as a faithful sim-

ple component, then FGeC(G,A, 1) is embedded in D and in particular

EF (G,A) = G, F (ζmk) ∗ G/A is a division algebra and D has degree at

least nk .


⟩, E = EF (G,A) and C ∈ CF (A). Then (A, 1) is a

strong Shoda pair of G and, by Theorem 2.1.6,

FGeC(G,A, 1) = M[G:E](F (ζmk) ∗ E/A)

is a Wedderburn component of FG which is a direct factor of the semisimple

algebra F ⊗QQGe(G,A, 1). It is easy to check that FGeC(G,A, 1) is a faithful

component of FG.

Assume now that FG contains as a faithful simple component a division

algebra D not isomorphic to FGeC(G,A, 1). Then also QG contains a division

algebra D′ not isomorphic to QGe(G,A, 1) as a faithful simple component.

By Lemma 3.2.12, QGe(G,A, 1) is embedded in D′. But this means that

FGeC(G,A, 1) is a direct factor of F ⊗Q QGe(G,A, 1), which in its turn is

embedded in F ⊗Q D′. If FGeC(G,A, 1) is not a division algebra, then it is a

matrix ring and it has nilpotent elements. But then also F ⊗QD′ has nilpotent

elements. Since F ⊗QD′ is a direct sum of isomorphic copies of D, also D has

nilpotent elements, which is a contradiction.

We conclude that FGeC(G,A, 1) is a division algebra of degree nk , which is

embedded in D. Necessarily E = G and D has degree at least nk .

81


Lemma 3.2.14

If N � G, (H, 1) is a strong Shoda pair of G and (HN/N, 1) is a strong

Shoda pair of G/N , then EF (G,H)N/N ⊆ EF (G/N,HN/N).

Proof This follows directly from the fact that if ζ|H| 7→ ζi|H| determines

an automorphism of Gal(F (ζ|H|)/F ), then it restricts to an automorphism

ζ[HN :N ] 7→ ζi[HN :N ] in Gal(F (ζ[HN :N ])/F ).

Because of the structure of metacyclic groups and their group algebras, we

can deduce that, for m a prime number, the above minimal faithful division

algebra component is essentially the only possible faithful division algebra

showing up in the Wedderburn decomposition.

Proposition 3.2.15

Let F be a number field, p a prime number and G = 〈a〉p ok 〈b〉n with

gcd(n, p) = 1. The only possible faithful division algebra components of

FG are the algebras FGeC(G,⟨abn/k

⟩, 1), with C ∈ CF (

⟨abn/k

⟩).

Proof Let b−1ab = ar. By Lemma 1.7.17, we deduce that the only non-

commutative components of FG are determined by the strong Shoda pairs

(Gd,K) with d a divisor of nk different from 1, Gd =

⟨a, bd

⟩, Gd/K cyclic and

d = min{x | nk : arx−1 ∈ K}. Let A = Gn/k =

⟨abn/k

⟩.

Assume that a strong Shoda pair (Gd,K) leads to a faithful division al-

gebra component of FG, then EF (G,Gd/K) = G and FGeC(G,Gd,K) =

F (ζ[Gd:K]) ∗ G/Gd has degree at least nk by Proposition 3.2.13. However its

degree equals |G/Gd| = d. Therefore d = nk , Gd = A and K =

⟨bln/k

⟩⊆ Z(G)

for l | k.

Since (A,K) is a Shoda pair and the characteristic subgroup K is con-

tained in the kernel of the character associated to eC(G,A,K), in order for

FGeC(G,A,K) to be a faithful component, K has to be 1.

Clearly the groups of type (Z)(a) are never F -critical because the groups

are abelian.

We study the groups of type (Z)(b). For this, we need two lemmas.

Lemma 3.2.16

Let F/Q be a finite normal extension such that F is totally real. Assume

ω = (ζm − ζ−1m )2 ∈ F for some positive integer m ≥ 3. Then ω is totally

negative.

82


Proof Let σ : F → R be an embedding of F in R. Since F/Q is normal,

σ(F ) = F . Also σ(ω) = −2 + σ(ζ2m + ζ−2

m ). Because F (ζm)/F is normal, the

map σ extends to an automorphism σ : F (ζm) → F (ζm) in the Galois group

of F (ζm)/Q. Therefore σ(ζm) = ζrm 6= 1, for some integer r, and σ(ζ2m + ζ−2

m )

is again a sum of 2 roots of unity. Hence |σ(ζ2m + ζ−2

m )| < 2 and σ(ω) < 0.

Lemma 3.2.17

Let F be a number field and n,m two positive integers such that n divides

m. If F splits(−1,(ζn−ζ−1

n )2

Q(ζn+ζ−1n )

), then F also splits

(−1,(ζm−ζ−1

m )2

Q(ζm+ζ−1m )

).

Proof Using the binomial formula and an induction argument, one easily

proves that ζkm + ζ−km ∈ Q(ζm + ζ−1m ). Therefore, Q(ζn + ζ−1

n ) ⊆ Q(ζm + ζ−1m ).

Also, for any positive integer k, the following equality is well known:

xk − yk = (x− y)(xk−1 + xk−2y + . . .+ xyk−2 + yk−1).

Applying this to ζn = ζkm, we have

ζn − ζ−1n = (ζm − ζ−1

m )(ζk−1m + ζk−3

m + . . .+ ζ−k+3m + ζ−k+1

m ). (11)

If F splits(−1,(ζn−ζ−1

n )2

Q(ζn+ζ−1n )

), then by Proposition 1.2.1 there exists a triple

(x, y, z) ∈ F (ζn + ζ−1n )3 \ {(0, 0, 0)} such that −x2 + (ζn − ζ−1

n )2y2 = z2. By

(11), it follows that the equation −x2 − (ζm − ζ−1m )2y2 = z2 holds for some

(x, y, z) ∈ F (ζm + ζ−1m )3 \ {(0, 0, 0)} and hence F splits

(−1,(ζm−ζ−1

m )2

Q(ζm+ζ−1m )

).

Theorem 3.2.18

Let F be an abelian number field and let G be a finite group. Then G

is a Z-group of type (Z)(b) and F -critical if and only if G = Cp o2 C4,

with p a prime number satisfying p ≡ −1 mod 4, F is totally imaginary,

Q(ζp) ∩ F ⊆ Q(ζp + ζ−1p ) and both ep(F/Q) and fp(F/Q) are odd.

In this case FG contains an exceptional component(−1,(ζp−ζ−1

p )2

F (ζp+ζ−1p )

).

Proof Let G = Cm o2 C4 =⟨a, b : am = 1, b4 = 1, b−1ab = a−1

⟩with m odd

and let F be an abelian number field. Assume that G is F -critical and let

A =⟨ab2⟩. Then EF (G,A) = G and FGeC(G,A, 1) =

(−1,(ζm−ζ−1

m )2

F (ζm+ζ−1m )

)is a

division algebra for any C ∈ CF (A) because of Lemma 3.2.11 and Proposi-

tion 3.2.13. By Lemma 3.2.10, EF (G,A) = G is equivalent with Q(ζm) ∩ F ⊆Q(ζm + ζ−1

m ).

83


Assume that m is not prime and choose a prime divisor p of m. Then G/N =

〈a〉p o2

⟨b⟩

4for N = 〈ap〉 � G. By Lemma 3.2.14, EF (G/N,A/N) = G/N .

Therefore, for any D ∈ CF (A/N), F (G/N)eD(G/N,A/N, 1) =(−1,(ζp−ζ−1

p )2

F (ζp+ζ−1p )

)is itself a division algebra by Lemma 3.2.17. Since G is F -critical, the algebra

F (G/N)eD(G/N,A/N, 1) has to be a totally definite quaternion algebra and

hence F (ζp + ζ−1p ) is totally real. But this means that F is totally real.

Since G is by assumption F -critical, one of the strong Shoda pairs (A,K)

with K ⊆ A and a /∈ K should produce a division algebra which is not a

totally definite quaternion algebra (see Lemma 1.7.17). The center of such

a FGeC(G,A,K) equals F (ζ[A:K] + ζ−1[A:K]) and is not totally real (since if

F (ζ[A:K] + ζ−1[A:K]) is totally real then −1 and (ζ[A:K] − ζ−1

[A:K])2 are totally

negative by Lemma 3.2.16). Therefore F is not totally real, a contradiction.

Hence F is totally imaginary, m = p prime and

FGeC(G,A, 1) = (F (ζp)/F (ζp + ζ−1p ), ζp 7→ ζ−1

p ,−1)

=

Ç−1, (ζp − ζ−1

p )2

F (ζp + ζ−1p )

å= F (ζp + ζ−1

p )⊗Q(ζp+ζ−1p )

Ç−1, (ζp − ζ−1

p )2

Q(ζp + ζ−1p )

åis a division algebra by Proposition 3.2.15. By Proposition 3.2.2 and Theo-

rem 1.5.3, this means that mp

(−1,(ζp−ζ−1

p )2

Q(ζp+ζ−1p )

)6= 1 and both the ramification

index ep(F (ζp + ζ−1p )/Q(ζp + ζ−1

p )) and fp(F (ζp + ζ−1p )/Q(ζp + ζ−1

p )) are odd.

By Lemma 1.5.5 and Example 1.4.11:

mp

Ç−1, (ζp − ζ−1

p )2

Q(ζp + ζ−1p )

å= min

®l ∈ N :

pf − 1


2

gcd(2, l)

´,

where e = ep(Q(ζp)/Q(ζp + ζ−1p )) = 2 and f = fp(Q(ζp + ζ−1

p )/Q) = 1.

We see that mp

(−1,(ζp−ζ−1

p )2

Q(ζp+ζ−1p )

)6= 1 if and only if p ≡ −1 mod 4. Also the

ramification index ep(Q(ζp + ζ−1p )/Q) = p−1

2 is odd since p ≡ −1 mod 4,

fp(Q(ζp + ζ−1p )/Q) = 1 and both ep(F (ζp + ζ−1

p )/F ) and fp(F (ζp + ζ−1p )/F )

divide p−12 which is odd. Therefore, by Theorem 1.4.4, the ramification index

ep(F (ζp+ζ−1p )/Q(ζp+ζ−1

p )) and the residue degree fp(F (ζp+ζ−1p )/Q(ζp+ζ−1

p ))

are odd if and only if ep(F/Q) and fp(F/Q) are odd. We conclude that G is

as in the statement of the theorem.

84


Assume now that G and F are as in the statement. Since p ≡ −1 mod 4,

C4 acts by inversion on Cp and clearly, G is a Z-group of type (Z)(b). Let

A =⟨ab2⟩. Since Q(ζp) ∩ F ⊆ Q(ζp + ζ−1

p ), EF (G,A) = G by Lemma 3.2.10.

Therefore FGeC(G,A, 1) =(−1,(ζp−ζ−1

p )2

F (ζp+ζ−1p )

)for any C ∈ CF (A). This sim-

ple component has degree 2 over its center F (ζp + ζ−1p ), which is totally

imaginary. Therefore it is not a totally definite quaternion algebra. Due

to, p ≡ −1 mod 4, mp

(−1,(ζp−ζ−1

p )2

Q(ζp+ζ−1p )

)6= 1 and because of the assumptions

both ep(F (ζp + ζ−1p )/Q(ζp + ζ−1

p )) and fp(F (ζp + ζ−1p )/Q(ζp + ζ−1

p )) are odd.

By Proposition 3.2.2, the simple component(−1,(ζp−ζ−1

p )2

F (ζp+ζ−1p )

)is now a division

algebra. Furthermore G has only quotients isomorphic to cyclic and dihedral

groups, which have only fields and matrix rings as simple components over F .

Therefore G is F -critical.

Lemma 3.2.19

Let G and H be finite groups with coprime order. If G×H is F -critical,

then G or H is a cyclic group.

Proof Assume that G × H is F -critical, then the simple division algebra

component equals F (G×H)e = FGe1⊗F FHe2 for some idempotents e, e1, e2

and both FGe1 and FHe2 are division algebras (including fields).

If both FGe1 and FHe2 are non-commutative division algebras, then at

least one of FGe1 and FHe2 is of odd degree because G and H have coprime

order. Therefore, they can not be both a totally definite quaternion algebra.

This means that G×H can never be F -critical, unless either G = 1 or H = 1.

If one of both, say FGe1, is a commutative division ring and thus a field,

then FGe1 is a Wedderburn component of F (G/G′). Because of the F -critical

condition, we can assume that G′ = 1 and hence G is an abelian group in (Z).

Since abelian groups in (Z) are cyclic, G is cyclic.

Theorem 3.2.20

Let F be an abelian number field and G a group. Then G is a Z-group of

type (Z)(c) and F -critical if and only if one of the following holds:

a) G = Cq×(Cpo2C4), with q and p different odd prime numbers with

oq(p) odd and p ≡ −1 mod 4. Moreover F is totally real and both

ep(F (ζq)/Q) and fp(F (ζq)/Q) are odd. In this case FG contains an

exceptional component(−1,(ζp−ζ−1

p )2


);

85


b) G = 〈a〉pok 〈b〉n, with n ≥ 8, p an odd prime number not dividing n,

b−1ab = ar, and both k and nk are divisible by all the prime numbers

dividing n. Moreover Q(ζp)∩F ⊆ Q(ζp+ζrp + . . .+ζrn/k−1

p ), mp = nk

and one of the following holds:

i. either p ≡ 1 mod 4 or n is odd, vq(p − 1) ≤ vq(k) for every

prime divisor q of n and mp,h <nk for every h 6= k divisor of k

such that vq(p − 1) ≤ vq(h) for every prime divisor q of n. If

n = 2k and F is totally imaginary, then mp,2 = 1.

ii. p ≡ −1 mod 4, v2(k) = 1, v2(n) = 2, vq(p − 1) ≤ vq(k) for

every odd prime divisor q of n and mp,h <nk for every h 6= k

divisor of k such that v2(h) = 1 and vq(p−1) ≤ vq(h) for every

odd prime divisor q of n.

iii. p ≡ −1 mod 4, v2(p + 1) + 1 ≤ v2(k), v2(n) = v2(k) + 1,

vq(p− 1) ≤ vq(k) for every odd prime divisor q of n and

(1) mp,h < nk for every divisor h of k different from k such

that v2(p+ 1) + 1 ≤ v2(h) and vq(p− 1) ≤ vq(k) for every

odd prime divisor q of n,

(2) mp,h <nk for every divisor h of k different from 2 and k

such that v2(h) = 1 and vq(p − 1) ≤ vq(h) for every odd

prime divisor q of n. If n = 2k and F is totally imaginary,

then mp,2 = 1.

Here

mp = min

®l ∈ N :

pf − 1


k

gcd(k, l)

´with K = F (ζk, ζp + ζrp + . . . + ζr

n/k−1

p ), e = ep(F (ζpk)/K) and

f = fp(K/Q), and

mp,h = min

®l ∈ N :

pfh − 1

gcd(pfh − 1, eh)≡ 0 mod

h

gcd(h, l)

´with Kh = K ∩ F (ζph), eh = ep(F (ζph)/Kh) and fh = fp(Kh/Q).

In this case, FG contains (K(ζp)/K, σ, ζk) as an exceptional compo-

nent, where the action is defined by

σ : F (ζpk)→ F (ζpk) : ζp 7→ ζrp ; ζk 7→ ζk.

86


Proof Assume that G is a Z-group of type (Z)(c) which is F -critical. Then

G = Cm0 okCn = Cm0 ok∏p|m0

Rp = Cm×(Pp1 okp1 Rp1)×· · ·×(Ppsokps Rps),

with k = kp1 · · · kps and p1, . . . , ps the prime divisors of m0 such that Cn does

not act trivially on Ppi . Here all pi are odd since p = 2 can never satisfy (3) of

Remark 1.10.2. By Lemma 3.2.19, necessarily G = Cm × (〈a〉pt ok 〈b〉n) with

p odd and gcd(m,n) = 1 = gcd(m, p) = gcd(n, p).

We claim that t = 1. Assume that t 6= 1 and let A = Cm ×⟨abn/k

⟩.

Then EF (G,A) = G and FGeC(G,A, 1) = F (ζmptk) ∗ G/A is a division al-

gebra for any C ∈ CF (A) because of Proposition 3.2.13 and Lemma 3.2.11.

Take N = 〈ap〉. Then G/N = Cm × (〈a〉p ok⟨b⟩n), EF (G/N,A/N) = G/N ,

by Lemma 3.2.14, and, for any D ∈ CF (A/N), F (G/N)eD(G/N,A/N, 1) =

F (ζmpk) ∗G/A is naturally embedded in FGeC(G,A, 1). Hence also F (G/N)

contains a simple component which is a division algebra of degree nk . Therefore

it should be a totally definite quaternion algebra and hence nk = 2. This means

that the action of⟨b⟩

on 〈a〉 is of order 2 and b−1ab = a−1. Since F (ζmk) is

contained in the center of F (ζmpk) ∗ G/A, which is totally real, m = 1 and

k ≤ 2. By Remark 1.10.2, k 6= 1. So k = 2 and n = 4. Furthermore, F is

totally real. Now G is as in case (Z)(b), but by Theorem 3.2.18, F cannot be

totally real in order for G to be F -critical. This contradiction tells us that

t = 1.

From now on G = Cm × (〈a〉p ok 〈b〉n) is F -critical and we distinguish two

cases, m 6= 1 and m = 1.

Assume first that m 6= 1 and G = Cm × (〈a〉p ok 〈b〉n) is F -critical. We

claim that m is a prime number. Let N = Cm and A = Cm ×⟨abn/k

⟩.

Then G/N = 〈a〉p ok⟨b⟩n, EF (G/N,A/N) = G/N by Lemma 3.2.14 and,

for any D ∈ CF (A/N), F (G/N)eD(G/N,A/N, 1) = F (ζpk) ∗ G/A. It is nat-

urally embedded in F (ζmpk) ∗ G/A and therefore it is again a division alge-

bra of degree nk . Therefore it should be a totally definite quaternion algebra

and hence nk = 2. As before, we conclude that k = 2, n = 4 and F is

totally real. Now for some normal subgroup M of G and some prime divi-

sor q of m, G/M = Cq × (〈a〉p o2

⟨b⟩

4). Then F (G/M)eD(G/M,A/M, 1) =

F (ζ2qp) ∗G/A =(−1,(ζp−ζ−1

p )2


)should be a totally definite quaternion alge-

bra, but its center contains F (ζq) which is not totally real. This is a contra-

diction and hence m is a prime number.

87


So we can assume that G = Cq × (〈a〉p o2 〈b〉4) with q and p different odd

prime numbers. Using the conditions of (Z)(c), we get 2 = 2o2(p) - oq(p).Hence oq(p) is odd. Moreover, if p ≡ 1 mod 4, then 1 = v2(k) ≥ v2(p−1) ≥ 2,

a contradiction. Thus p ≡ −1 mod 4. Now

FGeC(G,A, 1) = (F (ζqp)/F (ζq, ζp + ζ−1p ), ζp 7→ ζ−1

p ,−1)

=

Ç−1, (ζp − ζ−1

p )2

F (ζq, ζp + ζ−1p )

å= F (ζq, ζp + ζ−1

p )⊗Q(ζp+ζ−1p )

Ç−1, (ζp − ζ−1

p )2

Q(ζp + ζ−1p )

åis a division algebra. By Theorem 1.5.3 and Proposition 3.2.2, the local in-

dex mp

(−1,(ζp−ζ−1

p )2

Q(ζp+ζ−1p )

)6= 1 and both ep(F (ζq, ζp + ζ−1

p )/Q(ζp + ζ−1p )) and

fp(F (ζq, ζp + ζ−1p )/Q(ζp + ζ−1

p )) are odd.

By Example 1.4.11, ep(Q(ζp)/Q(ζp + ζ−1p )) = 2 and fp(Q(ζp + ζ−1

p )/Q) = 1,

hence it follows from Lemma 1.5.5 that mp

(−1,(ζp−ζ−1

p )2

Q(ζp+ζ−1p )

)6= 1 if and only

if p ≡ −1 mod 4. Since p ≡ −1 mod 4, [Q(ζp + ζ−1p ) : Q] is odd and

also ep(Q(ζp + ζ−1p )/Q), fp(Q(ζp + ζ−1

p )/Q), ep(F (ζq, ζp + ζ−1p )/F (ζq)) and

fp(F (ζq, ζp + ζ−1p )/F (ζq)) are odd. Therefore ep(F (ζq, ζp + ζ−1

p )/Q(ζp + ζ−1p ))

and fp(F (ζq, ζp + ζ−1p )/Q(ζp + ζ−1

p )) are odd if and only if ep(F (ζq)/Q) and

fp(F (ζq)/Q) are odd. This means that G and F are as in (a).

Conversely, assume that G and F are as in (a), then G = Cpq o2 C4 =

Cq × (〈a〉p o2 〈b〉4) is a Z-group of type (c). Let A = Cq ×⟨ab2⟩

and

C ∈ CF (A). Since F is totally real, clearly Q(ζp) ∩ F ⊆ Q(ζp + ζ−1p ) and

G = EF (G,A) by Lemma 3.2.10. Since FGeC(G,A, 1) has a totally imag-

inary center F (ζq, ζp + ζ−1p ), it is not a totally definite quaternion algebra.

Due to p ≡ −1 mod 4, mp

(−1,(ζp−ζ−1

p )2

Q(ζp+ζ−1p )

)6= 1 and because of the assumptions

both ep(F (ζq, ζp + ζ−1p )/Q(ζp + ζ−1

p )) and fp(F (ζq, ζp + ζ−1p )/Q(ζp + ζ−1

p )) are

odd. Because of Proposition 3.2.2, FGeC(G,A, 1) is now a division algebra.

Therefore FG contains an exceptional component of type EC1. The proper

non-abelian quotients of G are Cpo2C4, D2p, D2pq and, since F is totally real,

those groups give rise to simple components which are either fields, matrix

rings or totally definite quaternion algebras. Therefore G is F -critical.

88


Assume now that m = 1 and G =⟨a, b : ap = 1, bn = 1, b−1ab = ar

⟩is F -

critical, with p an odd prime number, gcd(p, n) = 1 and op(r) = nk . By

Remark 1.10.2, if q is a prime divisor of n then

1 ≤ vq(n

k) ≤ vq(p− 1) ≤ vq(k).

In particular vq(n) ≥ 2, so either n = 4 and G = Cp o2 C4 or n ≥ 8. How-

ever G = Cp o2 C4 is of type (Z)(b), thus we can assume that n ≥ 8. Let

A =⟨abn/k

⟩. By assumption, FG contains an exceptional component of type

EC1, so by Proposition 3.2.13, FGeC(G,A, 1) is a division algebra which has

degree nk , and in particular EF (G,A) = G. By Lemmas 3.2.10 and 1.4.13,

EF (G,A) = G is equivalent to Q(ζp)∩F ⊆ Q(ζp)〈r〉 = Q(ζp+ζrp+. . .+ζr

n/k−1

p ).

From the conditions on (Z)(c), if n is even, we have 2 ≤ v2(p − 1) ≤ v2(k)

when p ≡ 1 mod 4 and either v2(k) = 1 or 3 ≤ v2(p + 1) + 1 ≤ v2(k) when

p ≡ −1 mod 4. Due to nk divides p−1 (see Remark 1.10.2), if p ≡ −1 mod 4,

then v2(p− 1) = 1 and v2(n) = v2(k) + 1. Together with the other conditions

from (Z)(c) on the parameters of G, this gives rise to the following cases:

i. either p ≡ 1 mod 4 or n is odd, vq(p−1) ≤ vq(k) for every prime divisor

q of n;

ii. p ≡ −1 mod 4, v2(k) = 1, v2(n) = 2 and vq(p − 1) ≤ vq(k) for every

odd prime divisor q of n;

iii. p ≡ −1 mod 4, 3 ≤ v2(p + 1) + 1 ≤ v2(k), v2(n) = v2(k) + 1 and

vq(p− 1) ≤ vq(k) for every odd prime divisor q of n.

Let σ be the automorphism of F (ζpk) which maps ζp to ζrp and fixes F (ζk).

Let K = F (ζpk)〈σ〉. By our assumptions and Lemma 3.2.11, for any C ∈ CF (A)

we have FGeC(G,A, 1) = (K(ζp)/K, σ, ζk). Since FGeC(G,A, 1) is a division

algebra,

n

k= ind(FGeC(G,A, 1)) = lcm(mq(FGeC(G,A, 1)) : q = p,∞),

by Theorems 1.4.3 and 1.5.3. If m∞(FGeC(G,A, 1)) = 2, then K ⊆ R by

Lemma 1.5.4. Note that always ζk ∈ K, so k = 2. This implies that n is a

power of 2. However, when k = 2, G is in case (ii) and v2(n) = 2, a contra-

diction because of n ≥ 8. Thus, m∞(FGeC(G,A, 1)) = 1. By Lemma 1.5.5,

mp(FGeC(G,A, 1)) = mp as in the statement of the theorem. Therefore

89


nk = ind(FGeC(G,A, 1)) = mp and FGeC(G,A, 1) is an exceptional compo-

nent of FG (see Lemma 3.2.1).

For each of the cases (i)-(iii), let h | k, h 6= k be an integer satisfying the

conditions as in the statement of the theorem or h = 2 when k is even, and

let N =¨bhnk

∂⊆ Z(G). Then G/N = Cp oh Chn

kis a non-abelian proper

quotient of G. Note also that the prime divisors of n and hnk are the same

since the prime divisors of n and nk are the same. Since N ⊆ Z(G), the im-

ages of the actions of Cn on Cp and of Chnk

on Cp are the same. So each

Sylow q-subgroup of Chnk

acts non-trivial on Cp and G/N is again a Z-group

of type (Z)(b) or (Z)(c). By our assumption on G, G/N cannot have excep-

tional components. Therefore the simple component F (G/N)eD(G/N,A/N, 1)

of degree nk of F (G/N) is not exceptional for any D ∈ CF (A/N). Since

EF (G,A) = G, by Lemma 3.2.14, EF (G/N,A/N) = G/N , and then by

Lemma 3.2.11, F (G/N)eD(G/N,A/N, 1) = (Kh(ζp)/Kh, σ, ζh) with ζh ∈ Kh.

We also use σ to denote its restriction to F (ζph). Hence

ind(F (G/N)eD(G/N,A/N, 1))

= lcm(mq(F (G/N)eD(G/N,A/N, 1)) : q = p,∞) ≤ n

k

by Theorems 1.4.3 and 1.5.3. As we assume G is F -critical, the algebra

F (G/N)eD(G/N,A/N, 1) is either a matrix ring or a totally definite quater-

nion algebra. If F (G/N)eD(G/N,A/N, 1) is a totally definite quaternion al-

gebra, then the degree nk = 2 and Kh ⊆ R, so h = 2 and G/N = Cp o2 C4.

In this case F is totally real. If F (G/N)eD(G/N,A/N, 1) is a matrix ring,

then lcm(mq(F (G/N)eD(G/N,A/N, 1)) : q = p,∞) < nk . We claim that

m∞(F (G/N)eD(G/N,A/N, 1) = 1. Suppose that this local index equals 2,

then Kh ⊆ R by Lemma 1.5.4, thus h = 2. It follows that hnk = 4 and n

k = 2.

So

lcm(mq(F (G/N)eD(G/N,A/N, 1)) : q = p,∞) = 1,

and hence the local index m∞(F (G/N)eD(G/N,A/N, 1)) = 1, a contradic-

tion. Thus the local index m∞(F (G/N)eD(G/N,A/N, 1)) = 1 and hence

mp(F (G/N)eD(G/N,A/N, 1) = mp,h <nk , with mp,h the formula as in the

statement of the theorem because of Janusz’ formula in Lemma 1.5.5.

Suppose that h 6= 2, then Kh is not totally real and hence the algebra

F (G/N)eD(G/N,A/N, 1) is a matrix ring and mp,h < nk . If h = 2, then

n = 2k and G/N = Cp o2 C4. If F (G/N)eD(G/N,A/N, 1) is a matrix ring,

90


then mp,2 < 2, so mp,2 = 1. If F (G/N)eD(G/N,A/N, 1) is a totally definite

quaternion algebra, then F is totally real. Hence G is as in (b).

Conversely, assume now that G and F are as in (b). Since any prime divisor

q of n divides nk , the order of the image of the action of Cn on Cp, any Sy-

low q-subgroup of Cn acts non-trivial on Cp. Together with the assumptions

on n and k in (i)-(iii), this means that G is a Z-group of type (Z)(c). Let

A =⟨abn/k

⟩. By Lemma 3.2.10 and the assumptions on F , EF (G,A) = G.

Then by Lemma 3.2.11, FGeC(G,A, 1) = (K(ζp)/K, σ, ζk) for any C ∈ CF (A).

Furthermore, FGeC(G,A, 1) has degree nk and always ζk ∈ K. By Theo-

rems 1.4.3 and 1.5.3,

ind(FGeC(G,A, 1)) = lcm(mq(FGeC(G,A, 1)) : q = p,∞).

We claim that m∞(FGeC(G,A, 1)) = 1. Assume m∞(FGeC(G,A, 1)) = 2,

then K ⊆ R by Lemma 1.5.4, and hence k = 2 and n is a power of 2. So

G is as in (i) or (iii), but from both conditions we can deduce that k can-

not be 2. Thus, m∞(FGeC(G,A, 1)) = 1. Therefore ind(FGeC(G,A, 1)) =

mp(FGeC(G,A, 1)) and by Lemma 1.5.5 and the assumptions, it follows that

mp(FGeC(G,A, 1)) = mp = nk . By Lemma 3.2.1, FGeC(G,A, 1) is an excep-

tional component of type EC1.

In order to prove that G does not have proper quotients with exceptional

components of type EC1 in their Wedderburn decomposition over F , we ar-

gue by means of contradiction. Let N be a normal subgroup of G such that

F (G/N) contains an exceptional component of type EC1. Then G/N is non-

abelian and hence N =¨bhnk

∂⊆ Z(G) for some divisor h of k, h 6= k. Then

G/N = Cp oh Chnk

and without loss of generality we can assume that this

group contains a faithful exceptional component of type EC1. Then G/N is

as in (Z)(b) or (Z)(c), and by Proposition 3.2.15, F (G/N)eD(G/N,A/N, 1) =

(Kh(ζp)/Kh, σ, ζh) is an exceptional division algebra. Hence

n

k= ind(F (G/N)eD(G/N,A/N, 1)) =

lcm(mq(F (G/N)eD(G/N,A/N, 1)) : q = p,∞).

If G/N is as in (Z)(b), then G/N = Cpo2C4, h = 2 and n = 2k. So G is as

in (i) or (iii) and by the assumptions on G, F is totally real or mp,2 = 1. By

Lemma 3.2.11,

F (G/N)eD(G/N,A/N, 1) =

Ç−1, (ζp − ζ−1

p )2

F (ζp + ζ−1p )

å.

91


We assume that it is exceptional, so

lcm(mq(F (G/N)eD(G/N,A/N, 1)) : q = p,∞) = 2.

If F is totally real, then(−1,(ζp−ζ−1

p )2

F (ζp+ζ−1p )

)is a totally definite quaternion algebra,

a contradiction. Hence mp(F (G/N)eD(G/N,A/N, 1)) = mp,2 = 1, but then

m∞(F (G/N)eD(G/N,A/N, 1)) = 2 and F (G/N)eD(G/N,A/N, 1) is not an

exceptional component by Lemma 3.2.1, again a contradiction.

Hence G/N is as in (Z)(c). If m∞(F (G/N)eD(G/N,A/N, 1)) = 2, then

Kh ⊆ R by Lemma 1.5.4, and then h = 2. However, h = 2 implies that n

is a power of 2, p ≡ −1 mod 4 and v2(hnk ) = 2. It follows that hnk = 4 and

G/N = Cp o2 C4, a contradiction. So, m∞(F (G/N)eD(G/N,A/N, 1)) = 1.

Suppose that p ≡ 1 mod 4 or hnk is odd (equivalently n is odd; case (i)),

then vq(p − 1) ≤ vq(h) for all prime divisors q of hnk (which are exactly the

prime divisors of n). Assume first that h 6= 2, then by the assumptions,

mp(F (G/N)eD(G/N,A/N, 1)) = mp,h <nk , a contradiction. Now regard the

case when h = 2, then necessarily n is a power of 2 and G/N = Cp o2 C4, a

contradiction.

Assume that p ≡ −1 mod 4 and hnk is even (equivalently n is even). Then

vq(p − 1) ≤ vq(h) for all odd prime divisors q of n. Also, either v2(k) = 1

or v2(p + 1) + 1 ≤ v2(k). We first deal with v2(k) = 1 (case (ii)). In

this case, v2(n) = 2, v2(h) = 1 and h 6= 2 (as otherwise n = 4). So

mp(F (G/N)eD(G/N,A/N, 1)) < nk , a contradiction. Finally, suppose that

v2(p + 1) + 1 ≤ v2(k) (case (iii)). Then either v2(p + 1) + 1 ≤ v2(h) or

v2(h) = 1. If h 6= 2, then mp(F (G/N)eD(G/N,A/N, 1)) < nk , a contradiction.

When h = 2, G/N = Cp o2 C4 gives again a contradiction.

We conclude that G does not have proper quotients with exceptional com-

ponents of type EC1, hence G is F -critical.

We summarize our results on F -critical groups.

Theorem 3.2.21

Let D be a division ring and F an abelian number field, p and q different

odd prime numbers. Then D is a Wedderburn component of FG for an

F -critical group G if and only if one of the following holds:

(a) D =(−1,−1

F

), G ∈ {SL(2, 3), Q8}, F is totally imaginary and both,

e2(F/Q) and f2(F/Q), are odd;

92

3.3 examples

(b) D =Ä−1,−1F (ζp)

ä, G ∈ {SL(2, 3)×Cp, Q8×Cp}, gcd(p, |G|/p) = 1, op(2)

is odd, F is totally real and both, e2(F (ζp)/Q) and f2(F (ζp)/Q), are

odd;

(c) D =(−1,(ζp−ζ−1

p )2

F (ζp+ζ−1p )

), G = Cp o2 C4, p ≡ −1 mod 4, F totally

imaginary, Q(ζp)∩F ⊆ Q(ζp+ζ−1p ) and both, ep(F/Q) and fp(F/Q),

are odd;

(d) D =(−1,(ζp−ζ−1

p )2


), G = Cq × (Cp o2 C4), p ≡ −1 mod 4, oq(p)

odd, F is totally real and both, ep(F (ζq)/Q) and fp(F (ζq)/Q), are

odd;

(e) D = (K(ζp)/K, σ, ζk) with Schur index nk , G = 〈a〉p ok 〈b〉n with

n ≥ 8, gcd(p, n) = 1, b−1ab = ar, and both k and nk are divisible

by all the primes dividing n. Here K = F (ζk, ζp + ζrp + ...+ ζrnk−1

p )

and σ : F (ζpk)→ F (ζpk) : ζp 7→ ζrp ; ζk 7→ ζk. Moreover Q(ζp) ∩ F ⊆Q(ζp + ζrp + ... + ζr

nk−1

p ) and one of the conditions (i) - (iii) from

Theorem 3.2.20 holds. Furthermore

min

®l ∈ N

∣∣∣∣ pf − 1


k

gcd(k, l)

´=n

k

with e = ep(F (ζpk)/K) and f = fp(K/Q).

Proof This follows by combining the results from Theorems 1.10.1, 3.2.18

and 3.2.20 and Propositions 3.2.4 to 3.2.7 and 3.2.9.

3.3 examples

We apply our results to the case when F is Q. This is a restatement of the

main result from [CdR14].

Corollary 3.3.1 (Caicedo-del Rıo)

Let G be a finite group.

1. QG contains an exceptional component of type EC1 if and only if

G contains an epimorphic image H and one of the following holds:

93


a) H = Cq×(Cpo2C4), with p and q different odd prime numbers,

oq(p) is odd and p ≡ −1 mod 4;

b) H = 〈a〉p ok 〈b〉n, with n ≥ 8, p an odd prime number not

dividing n, both k and nk are divisible by all the prime numbers

dividing n and one of the following holds:

i. k = gcd(n, p− 1) and either p ≡ 1 mod 4 or n is odd;

ii. k = gcd(n, p− 1), p ≡ −1 mod 4 and v2(n) = 2;

iii. p ≡ −1 mod 4, n = 2v2(p+1)+2 and k = 2v2(p+1)+1;

c) H = Q8 × Cp, with p an odd prime number and op(2) odd;

d) H = SL(2, 3)×Cp, with p a prime number different from 2 and

3 and op(2) odd;

2. QG contains an exceptional component B of type EC2 if and only

if G has an epimorphic image H such that the line H, Q, B appears

in Table 2.

Moreover, QG has exactly the following exceptional components of type

EC1:

�

(−1,(ζp−ζ−1

p )2

Q(ζq,ζp+ζ−1p )

)if G has an epimorphic image in case (a);

� (Q(ζkp)/Q(ζk, ζp+ζrp+. . .+ζrn/k−1

p ), σ, ζk), with r such that b−1ab =

ar and σ : ζp 7→ ζrp , if G has an epimorphic image in case (b);

�

Ä−1,−1Q(ζp)

äif G has an epimorphic image in case (c) or (d).

Proof This is readily verified using Theorems 3.1.2 and 3.2.21, except for the

fact why (i), (ii), (iii) is equivalent with (i), (ii), (iii) from Theorem 3.2.20.

We use the notations from Theorem 3.2.20. First note that f = fp(K/Q) =

ok(p) and e = ep(Q(ζpk)/K) = op(r) = nk for K = Q(ζk, ζp+ζrp + . . .+ζr

n/k−1

p )

by Theorem 1.4.9.

Assume that either p ≡ 1 mod 4 or n is odd. We prove that k = gcd(n, p−1)

if and only if mp = nk , vq(p − 1) ≤ vq(k) for every prime divisor q of n and

mp,h <nk for every h 6= k divisor of k such that vq(p − 1) ≤ vq(h) for every

prime divisor q of n. Suppose that k = gcd(n, p−1), then gcd(nk ,p−1k ) = 1 and

since each prime divisor of n also divides nk , also gcd(n, p−1

k ) = 1. Therefore,

94

3.3 examples

vq(p − 1) = vq(k) for every prime divisor q of n. Also ok(p) = 1 and hence

mp = min{l ∈ N : kgcd(k,l) |

p−1k

kgcd(nk ,p−1)} = n

k . Since vq(p − 1) = vq(k), for

each prime divisor q of n, there does not exist any proper divisor h of k such

that vq(p− 1) ≤ vq(h), so we do not have to prove anything for those divisors.

Conversely, since vq(p− 1) ≤ vq(k) for every prime divisor q of n, it already

follows that gcd(n, p − 1) | k. Assume now that k 6= gcd(n, p − 1). Then

h = gcd(n, p − 1) is a proper divisor of k with vq(p − 1) = vq(h) for every

prime divisor q of n. By the above, also mp,h = nk , a contradiction.

Assume now that p ≡ −1 mod 4 and v2(n) = 2. Using the same arguments

as in the previous case, we prove that k = gcd(n, p− 1) if and only if mp = nk ,

vq(p − 1) ≤ vq(k) for every odd prime divisor q of n and mp,h <nk for every

h 6= k divisor of k such that v2(h) = 1 and vq(p − 1) ≤ vq(h) for every odd

prime divisor q of n.

At last, assume that p ≡ −1 mod 4. We prove that n = 2v2(p+1)+2 and

k = 2v2(p+1)+1 if and only if mp = nk , v2(p+ 1) + 1 ≤ v2(k), v2(n) = v2(k) + 1,

vq(p− 1) ≤ vq(k) for every odd prime divisor q of n and

(1) mp,h <nk for every divisor h of k different from k such that v2(p+1)+1 ≤

v2(h) and vq(p− 1) ≤ vq(k) for every odd prime divisor q of n,

(2) mp,h < nk for every divisor h of k different from 2 and k such that

v2(h) = 1 and vq(p− 1) ≤ vq(h) for every odd prime divisor q of n.

Assume that n = 2v2(p+1)+2 and k = 2v2(p+1)+1, then clearly v2(p+1)+1 =

v2(k), v2(n) = v2(k) + 1, vq(p−1) ≤ vq(k) for every odd (none!) prime divisor

q of n. Because of the structure of U(Z/kZ), we know that ok(p) = v2(p + 1)

and 2k - pok(p) − 1. Therefore mp = min{l ∈ N : kgcd(k,l) |

pok(p)−12 } = 2 = n

k .

Since v2(p + 1) + 1 = v2(k), there does not exist any proper divisor h of k

such that v2(p+ 1) + 1 ≤ v2(h), so we do not have to prove anything for those

divisors. Also, if v2(h) = 1, then h = 2, so we are finished.

Conversely, we claim that v2(k) = v2(p + 1) + 1 and vq(k) = vq(p − 1)

for every odd prime divisor q of n. Assume first that v2(k) > v2(p + 1) + 1,

then there exists a proper divisor h of k with v2(h) = v2(p + 1) + 1 and

vq(h) = vq(p − 1). One computes that mp,h = nk , a contradiction by (1).

Assume now that vq(k) > vq(p− 1) for some prime divisor q of n. Then there

exists a proper prime divisor h of k with v2(h) = 1 and vq(p − 1) = vq(h).

Hence h = gcd(p − 1, n) and by the above mp,h = nk . By (2), it follows that

h = 2. But then nk = mp,h ≤ h = 2 and k is a power of 2, a contradiction.

95


So, v2(k) = v2(p+ 1) + 1 and vq(k) = vq(p− 1) for every odd prime divisor

q of n. Set n = 2v2(p+1)+2n1 with 2 - n1. Then k = 2v2(p+1)+1 gcd(n1, p − 1).

Suppose that gcd(n1, p− 1) 6= 1, then h = 2 gcd(n1, p− 1) = gcd(n, p− 1) is a

proper divisor of k different from 2. Moreover, v2(h) = 1 and vq(p−1) ≤ vq(h)

for every odd prime divisor q of n. By the above, mp,h = nk , which contradicts

(2). Hence gcd(n1, p − 1) = 1. Since nk divides p − 1 and both k and n

k are

divisible by all the prime numbers dividing n, necessarily n = 1 and the result

follows.

The conditions of Theorem 3.2.21 are easy to check algorithmically and we

did implement it in GAP. With this program we can compute the F -critical

groups for any abelian number field F up to a fixed order. As an illustration

we include the F -critical groups up to order 200 for all subfields of Q(ζ7). We

compute the Schur index of the corresponding exceptional component A and

we denote the center of A in the standard GAP notation. A local Schur index

[p, s] means that mp(A) = s and mq(A) = 1 for all other prime numbers

q. When for a fixed group, there are multiple lines in the table, this means

that the exceptional component A appears as several isomorphic copies in the

Wedderburn decomposition.

Table 3: List of Q(ζ7)-critical groups of type EC1 up to order 200

ID Structure Center Schur index Local index

[8, 4] Q8 CF(7) 2 [2, 2]

[24, 3] SL(2, 3) CF(7) 2 [2, 2]

[44, 1] C11 o2 C4 NF(77,[ 1, 43 ]) 2 [11, 2]

[48, 1] C3 o8 C16 CF(56) 2 [3, 2]

[80, 1] C5 o8 C16 NF(280,[ 1, 169 ]) 2 [5, 2]

[92, 1] C23 o2 C4 NF(161,[ 1, 22 ]) 2 [23, 2]

[117, 1] C13 o3 C9 NF(273,[ 1, 22, 211 ]) 3 [13, 3]

[160, 3] C5 o8 C32 CF(56) 4 [5, 4]

[172, 1] C43 o2 C4 NF(301,[ 1, 85 ]) 2 [43, 2]

96

3.3 examples

Table 4: List of Q-critical groups of type EC1 up to order 200

ID Structure Center Schur index Local index

[40, 1] C5 o4 C8 NF(20,[ 1, 9 ]) 2 [5, 2]

[48, 1] C3 o8 C16 CF(8) 2 [3, 2]

[56, 10] C7 ×Q8 CF(7) 2 [2, 2]

[63, 1] C7 o3 C9 NF(21,[ 1, 4, 16 ]) 3 [7, 3]

[80, 3] C5 o4 C16 GaussianRationals 4 [5, 4]

[84, 4] C3 × (C7 o2 C4) NF(21,[ 1, 13 ]) 2 [7, 2]

[104, 1] C13 o4 C8 NF(52,[ 1, 25 ]) 2 [13, 2]

[117, 1] C13 o3 C9 NF(39,[ 1, 16, 22 ]) 3 [13, 3]

[132, 1] C11 × (C3 o2 C4) CF(11) 2 [3, 2]

[156, 3] C13 × (C3 o2 C4) CF(13) 2 [3, 2]

[168, 22] C7 × SL(2, 3) CF(7) 2 [2, 2]

[176, 1] C11 o8 C16 NF(88,[ 1, 65 ]) 2 [11, 2]

[184, 10] C23 ×Q8 CF(23) 2 [2, 2]

97


Table

5:List

ofQ

(ζ7+ζ−1

7)-critica

lgroupsoftypeEC1upto

order

200

IDStru

cture

Center

Schurindex

Localindex

[40,1]

C5o

4C

8NF(140,[1,29,41,69])

2[5,2]

[48,1]

C3o

8C

16

NF(56,[1,41])

2[3,2]

[56,10]

C7×Q

8CF(7)

2[2,2]

CF(7)

2[2,2]

CF(7)

2[2,2]

[80,3]

C5o

4C

16

NF(28,[1,13])

4[5,4]

[84,4]

C3×

(C7o

2C

4 )NF(21,[1,13])

2[7,2]

NF(21,[1,13])

2[7,2]

NF(21,[1,13])

2[7,2]

[104,1]

C13o

4C

8NF(364,[1,181,209,337])

2[13,2]

[117,1]

C13o

3C

9NF(273,[1,22,55,118,139,211])

3[13,3]

[132,1]

C11×

(C3o

2C

4 )NF(77,[1,34])

2[3,2]

[156,3]

C13×

(C3o

2C

4 )NF(91,[1,27])

2[3,2]

[168,22]

C7×

SL(2,3)

CF(7)

2[2,2]

CF(7)

2[2,2]

CF(7)

2[2,2]

[176,1]

C11o

8C

16

NF(616,[1,153,265,505])

2[11,2]

[184,10]

C23×Q

8NF(161,[1,139])

2[2,2]

98

3.3 examples

Table

6:ListofQ(√−7)-criticalgroupsoftypeEC1upto

order

200

IDStructure

Center

Schurindex

Localindex

[8,4]

Q8

NF(7,[1,2,4])

2[2,2]

[24,3]

SL(2,3)

NF(7,[1,2,4])

2[2,2]

[44,1]

C11o

2C

4NF(77,[1,23,32,43,65,67])

2[11,2]

[48,1]

C3o

8C

16

NF(56,[1,9,25])

2[3,2]

[63,1]

C7o

3C

9NF(21,[1,4,16])

3[7,3]

NF(21,[1,4,16])

3[7,3]

[80,1]

C5o

8C

16

NF(280,[1,9,81,121,169,249])

2[5,2]

[92,1]

C23o

2C

4NF(161,[1,22,93,114,116,137])

2[23,2]

[117,1]

C13o

3C

9NF(273,[1,16,22,79,100,172,211,235,256])

3[13,3]

[160,3]

C5o

8C

32

NF(56,[1,9,25])

4[5,4]

[172,1]

C43o

2C

4NF(301,[1,44,85,128,130,214])

2[43,2]

99

4C E N T R A L U N I T S

In this chapter, we study Z(U(ZG)), the group of central units of ZG for finite

groups G.

Let Γ be a finitely generated abelian group. Assume that Γ has rank r

and let u1, . . . , ur ∈ Γ. Then 〈u1, . . . , ur〉 has finite index in Γ if and only if

u1, . . . , ur are multiplicatively independent. A virtual basis of Γ is a set of

multiplicatively independent elements of Γ which generate a subgroup of finite

index in Γ.

Siegel, Borel and Harish-Chandra showed that Z(U(ZG)) is finitely gener-

ated (Theorem 1.8.7) and its rank is known (Theorem 1.14.2). Therefore, one

knows that a virtual basis of Z(U(ZG)) exists and one knows the number of

elements in such a basis.

In this chapter, we construct a virtual basis of Z(U(ZG)) for three classes

of finite groups: for finite abelian groups; for finite abelian-by-supersolvable

groups such that every cyclic subgroup of order not a divisor of 4 or 6, is

subnormal in G; and for finite strongly monomial groups such that there exists

a complete and non-redundant set of strong Shoda pairs (H,K) of G, with the

property that each [H : K] is a power of a prime number.

4.1 abelian groups

In this section, we prove the Bass-Milnor Theorem (Theorem 1.12.1): the

group generated by the Bass units is of finite index in U(ZG), for G a finite

abelian group. We give a constructive proof that hence also provides insight in

the techniques needed and difficulties one encounters in order to find finitely

many generators for a subgroup of finite index. Additionally, we also discover

a virtual basis consisting of Bass units.

101

central units

We assume that G is a finite abelian group. We know already that U(ZG)

is finitely generated abelian (Theorem 1.8.7) and has rank r = 1+k2+|G|−2c2 ,

where c is the number of cyclic subgroups of G and k2 is the number of elements

of G of order 2 (Theorem 1.14.3). Bass took a concrete list of r Bass units

of ZG for G cyclic and proved that they are multiplicatively independent. To

do so, he used the Bass Independence Theorem which in turn uses the Franz

Independence Lemma (see [Seh93, Lemma 11.3, Theorem 11.8] for details).

Bass and Milnor proved, using K-Theory, that the group generated by the

units of integral group rings of cyclic subgroups of G has finite index in U(ZG).

However, their proof is not constructive.

Alternatively, assume that Γ is a subgroup of finite index in a finitely gen-

erated abelian group Λ and that we know a subset X of Λ which generates a

subgroup of finite index in Λ. Let Y be a subset of Γ. Then 〈Y 〉 has finite

index in Γ if and only if for every x ∈ X there is a positive integer m such

that xm ∈ 〈Y 〉. In our proof we take Γ = U(ZG), Y the set of Bass units of

ZG, Λ = U(O), where O is the unique maximal order of QG, and X the set

of cyclotomic units of QG.

By Theorem 1.6.4 of Perlis and Walker, QG is isomorphic to a direct product

of cyclotomic fields. For simplicity, we consider this isomorphism as an equality

QG =⊕k

i=1 Q(ζni). Then ZG is an order of QG and O =⊕k

i=1 Z[ζni ] is the

unique maximal order of QG. In particular, ZG ⊆ O. Moreover, Γ = U(ZG)

has finite index in Λ = U(O) (see Lemma 1.8.6). Having in mind that the

cyclotomic units of Q(ζni) generate a subgroup of finite index in U(Z[ζni ]) (see

Section 1.11), we conclude that the set X of cyclotomic units of QG generates

a subgroup of finite index in Λ. Thus the Bass-Milnor Theorem is equivalent

to the following proposition.

Proposition 4.1.1

Let G be a finite abelian group and let QG =⊕k

i=1 Q(ζni), the realization

of the Perlis-Walker Theorem. Then for every cyclotomic unit u of QGthere is a positive integer m such that um is a product of Bass units of

ZG.

The proof of Proposition 4.1.1 (actually of Lemma 4.1.3) is constructive and

avoids K-theory and independence arguments. This is the first result of this

section. The second result consists in giving a concrete virtual basis B formed

by Bass units for U(ZG).

102

4.1 abelian groups

4.1.1 A new proof of the Bass-Milnor Theorem

Throughout the rest of the section G is a finite abelian group. In this section

we prove Proposition 4.1.1. This provides a new proof of the Bass-Milnor

Theorem which states that B1(G) has finite index in U(ZG).

First of all we obtain a precise realization of the Perlis-Walker Theorem

which states that there is an isomorphism f : QG →⊕

dQ(ζd)kd , where kd

denotes the number of cyclic subgroups of G of order d. This isomorphism is

realized as follows. Let H = H(G) denote the set of subgroups H of G such

that G/H is cyclic. For every subgroup H ∈ H, we fix a linear representation

ρH of G with kernel H. We also denote by ρH the linear extension of ρH to

QG. If d = [G : H] then ρH(QG) = Q(ζd), where ζd denotes a primitive d-th

root of unity. Then

f =⊕H∈H

ρH : QG −→⊕H∈H

Q(ζ[G:H])

is an isomorphism of algebras. This isomorphism is the same as the one of

Perlis and Walker in Corollary 1.6.5, since kd equals the number of subgroups

H ∈ H such that [G : H] = d.

The following equalities are easy to check:

n−1∏i=0

(1−Xζin) = 1−Xn, (12)

ρH(uk,m(g)) = ηk(ρH(g))m, (13)

where H ∈ H, g ∈ G and km ≡ 1 mod |g|.Let ξ be a root of unity and assume that k is coprime to n and the order of

ξ. If ξn 6= 1 then, using equation (12), we obtain

n−1∏i=0

ηk(ξζin) =n−1∏i=0

1− ξkζkin1− ξζni

=

∏n−1i=0 (1− ξkζin)∏n−1i=0 (1− ξζni)

=1− ξkn

1− ξn= ηk(ξn).

Otherwise, i.e. if ξn = 1, then ξζjn = 1 for some j = 0, 1, . . . , n − 1. Then,

using that k is coprime to n, we deduce that

n−1∏i=0

ηk(ξζin) =n−1∏

i=0,i6=j

1− ζk(i−j)n

1− ζi−jn

=

∏n−1i=1 (1− ζkin )∏n−1i=1 (1− ζin)

= 1 = ηk(ξn).

103

central units

This proves the following equality for every primitive l-th root of unity ξ:

n−1∏i=0

ηk(ξζin) = ηk(ξn) (gcd(k, nl) = 1). (14)

Lemma 4.1.2

Let g ∈ G, H ∈ H and K be an arbitrary subgroup of G. Let h = |H∩K|,t = [K : H∩K] and let k and m be positive integers such that gcd(k, t) = 1

and km ≡ 1 mod |gu| for every u ∈ K. Then∏u∈K

ρH(uk,m(gu)) = ηk(ρH(g)t)mh.

Proof As H = ker(ρH), if u runs through the elements of K then ρH(u) runs

through the t-th roots of unity and each t-th root of unity is obtained as ρH(u)

for precisely h elements u of K. Therefore

∏u∈K

ρH(uk,m(gu)) =

(∏u∈K

ηk(ρH(g)ρH(u))

)m

=

(t−1∏i=0

ηk(ρH(g)ζit)

)mh= ηk(ρH(g)t)mh

as desired. We have used equation (14) in the last equality.

Proposition 4.1.1 is a consequence of the following stronger lemma.

Lemma 4.1.3

Let H ∈ H with d = [G : H] and let k, j ∈ N be such that k is coprime to

d. Set η = ηk(ζjd) and let B1,k(G) be the subgroup of U(ZG) generated by

the Bass cyclic units of the form uk,l(g) with g ∈ G and kl ≡ 1 mod |g|.Then there is a positive integer m and b ∈ B1,k(G) such that ρH(b) = ηm

and ρK(b) = 1 for every K ∈ H \ {H}.

Proof Without loss of generality, we may assume that k is coprime to n = |G|.Indeed, by an easy Chinese Remainder argument there is an integer k′ coprime

to n such that k ≡ k′ mod d. Then clearly ηk(ζjd) = ηk′(ζjd).

104

4.1 abelian groups

We argue by a double induction, first on n and second on d. The cases n = 1

and d = 1 are trivial. We denote by P (G,H) the statement of the lemma for

a finite abelian group G and an H ∈ H(G). Hence the induction hypothesis

includes the following statements:

(IH1): P (M,Y ) holds for every proper subgroup M of G and any Y ∈ H(M).

(IH2): P (G,H1) holds for every H1 ∈ H(G) with [G : H1] < [G : H] = d.

We consider two cases, depending on whether j is coprime to d or not.

Case 1: j is not coprime to d. Let p be a common prime divisor of d and j.

Then H is contained in a subgroup S of G with [G : S] = p and ζjd = ζj/pd′ with

d′ = [S : H]. For every K ∈ H(G), let λK denote the restriction of ρK to QS.

Clearly λK is the Q-linear extension of a linear representation of S with kernel

S∩K. Since S/(S∩K) ' KS/K and KS/K is a subgroup of G/K we deduce

that S/(S ∩ K) is cyclic. Thus K → S ∩ K defines a map H(G) → H(S).

This map is surjective, but maybe not injective. Indeed, let K1 ∈ H(S). If

K1 ∈ H(G) then clearly the map associates K1 with K1. Otherwise p divides

[G : K1] and S/K1 is a cyclic subgroup of G/K1 of maximal order. This implies

that G/K1 = S/K1 × L/K1 for some subgroup L of G containing K1 and so

that [L : K1] = p. Then G/L ' S/K1, so that L ∈ H(G), and L ∩ S = K1.

Therefore H(S) = {K ∩ S : K ∈ H(G)}. For every Y ∈ H(S) we choose a

KY ∈ H(G) such that KY ∩ S = Y in such a way that KY = Y if Y ∈ H(G).

Then ⊕Y ∈H(S)

λKY : QS →⊕

Y ∈H(S)

Q(ζ[S:Y ])

is an algebra isomorphism. By the first induction hypothesis (IH1) there is

an element b ∈ B1,k(S) such that λH(b) = ηk(ζj/pd′ )m = ηm for some positive

integer m and ρK(b) = λK(b) = 1 if K ∈ H(G) with K ∩S 6= H. If K ∈ H(G)

satisfies K ∩ S = H then either K = H or K = H1, where H1/H ' G/S is

the only subgroup of G/H of order p, since G/H is cyclic. Moreover ρH1(b)

is a product of cyclotomic units of Q(ζd), by equation (13). By the second

induction hypothesis (IH2) there is c ∈ B1,k(G) such that ρK(c) = 1 for

every K ∈ H(G) \ {H1} and ρH1(c) = ρH1

(b)m1 for some positive integer m1.

Therefore ρH(bm1c−1) = ηmm1 and ρK(bm1c−1) = 1 for every K ∈ H(G)\{H}.This finishes the proof for this case.

105

central units

Case 2: j is coprime to d. Then G = 〈a,H〉 and ρH(a) = ζjd for some a ∈ G.

As k is coprime to n, there is a positive integer m such that km ≡ 1 mod |au|for every u ∈ H. Hence

ηm = ρH(uk,m(a)),

by equation (13). Let

b =∏h∈H

uk,m(ah).

For every K ∈ H(G), set

dK = [G : K], d′K = [G : 〈a,K〉], hK = |H ∩K| and tK = [H : H ∩K].

Then, by Lemma 4.1.2,

ρK(b) = ηk(ρK(a)tK )mhK = ηk(ζtKd

′KuK

dK)mhK ,

for some integer uK coprime to dK . If tKd′K is not coprime to dK then, by

Case 1, there is bK ∈ B1,k(G) such that ρK(bK) = ρK(b)mK for some integer

mK and ρK1(bK) = 1 for K1 ∈ H(G) \ {K}. By (IH2), the same holds if

dK < d. Let

H′ = {K ∈ H(G) : tKd′K is not coprime to dK or dK < d}.

For each K ∈ H′ fix bK ∈ B1,k(G) and mK ∈ N as above and denote m1 =

lcm(mK : K ∈ H′) and

b1 =

( ∏K∈H′

b− m1mK

K

)bm1 .

Then b1 ∈ B1,k(G), ρK(b1) = 1 if K ∈ H′ and ρK(b1) ∈ 〈ηk(ρK(a)tK )〉 if

K ∈ H(G) \ H′. Observe that tH = d′H = uH = 1 and dH = d and hence

H 6∈ H′. Therefore, ρH(b1) ∈ 〈η〉, because ηk(ρH(a)tH ) = η. To finish the

proof we show that H′ = H(G) \ {H}. Suppose the contrary, that is, assume

K ∈ H(G)\{H} with dK ≥ d and gcd(tKd′K , dK) = 1. The latter implies that

d′K = 1, or equivalently G = 〈a,K〉, and tK = [KH : K] is coprime to dK =

[G : K]. Consequently, tK = 1, or equivalently H ⊆ K. Hence, the assumption

dK = [G : K] ≥ [G : H] = d implies that H = K, a contradiction.

Note that the proof of Lemma 4.1.3 provides a recursive algorithm that for

a cyclotomic unit η of QG as input, returns an integer m and an expression of

ηm as a product of Bass units of ZG. For more details, see [JdRVG14].

106

4.1 abelian groups

As it was mentioned in the introduction of this section, the Bass-Milnor

Theorem is equivalent to Proposition 4.1.1 and the well known fact that the

cyclotomic units of Q(ζn) generate a subgroup of finite index of U(Z[ζn]) for

every root of unity ζn. We include a proof for completeness.

Theorem 4.1.4 (Bass-Milnor)

If G is a finite abelian group then B1(G) has finite index in U(ZG).

Proof Let H = H(G), f =⊕

H∈H ρH : QG →⊕

H∈HQ(ζ[G:H]) and V be

the subgroup of⊕

H∈H U(Z(ζ[G:H])) generated by the cyclotomic units of QG,

i.e. generated by the units that project on one component to a cyclotomic

unit and project on all other components to 1. As the cyclotomic units of each

ring Q(ζn) generate a subgroup of finite index in U(Z[ζn]), V has finite index

in⊕

H∈H U(Z(ζ[G:H])). Hence, by Lemma 4.1.3, f(B1(G)) has finite index

in⊕

H∈H U(Z(ζ[G:H])) and therefore B1(G) has finite index in U(ZG), since

B1(G) ⊆ U(ZG) ⊆ f−1(⊕

H∈H U(Z(ζ[G:H]))).

4.1.2 A virtual basis of Bass units

Bass proved that if G = 〈g〉, a cyclic group of order n, and m is a multiple

of φ(n) then{uk,m(gd) : 1 < k < n

2d , (k,nd ) = 1

}is a virtual basis of U(ZG)

[Bas66]. In this section we generalize this result and obtain a virtual basis

consisting of Bass units for the unit group of the integral group ring of an

arbitrary abelian group G. Moreover, the proof provides, for an arbitrary

Bass unit b, an algorithm to express a power of b as a product of a trivial unit

and powers of at most two units in this basis of Bass units.

Theorem 4.1.5




|C|2, gcd(k, |C|) = 1

™is a virtual basis of U(ZG).

Proof The proof is based on the equalities from Lemma 1.12.2.

107

central units

Let C be the set of cyclic subgroups of G. By Theorem 4.1.4, B1(G) is a

subgroup of finite index in U(ZG). Let t = φ(|G|). We first prove that

B1 =

ßuk,t(aC) : C ∈ C, 1 < k <

|C|2, gcd(k, |C|) = 1

™generates a subgroup of finite index in U(ZG). To do so we “sieve” gradually

the list of Bass units, keeping the property that the remaining Bass units still

generate a subgroup of finite index in U(ZG), until the remaining Bass units are

the elements of B1. By equation (1), to generate B1(G) it is enough to use the

Bass units of the form uk,m(g) with g ∈ G, 1 ≤ k < |g| and km ≡ 1 mod |g|.By equation (6), for every Bass unit uk,m(g) we have uk,m(g)u = uk,t(g)v

for some positive integers u and v. Thus the Bass units of the form uk,t(g)

with 1 ≤ k < |g| and gcd(k, |g|) = 1 generate a subgroup of finite index in

U(ZG). By equations (2) and (3), we can reduce further the list of generators

by taking only those with g = aC for some cyclic group C of G. By equations

(4) and (5) we can exclude the Bass units with k = ±1 and still generate

a subgroup of finite index in U(ZG) with the remaining elements. Finally,

uk,t(g)−1u|g|−k,t(g) = u|g|−1,t(gk) = (−gk)−t, by equations (3), (5) and (7).

Thus uk,t(g)−1u|g|−k,t(g) has finite order. Therefore the units uk,t(g) with

k > |g|2 can be excluded. The remaining units are exactly the elements of B1.

Thus 〈B1〉 has finite index in U(ZG), as desired.

Let B = {uk,mk,C (aC) : C ∈ C, 1 < k < C2 , gcd(k, |C|) = 1}. Using equation

(6) once more, we deduce that 〈B〉 has finite index in U(ZG), since so does

〈B1〉.To finish the proof we need to show that the elements of B are multiplica-

tively independent. To do so, it is enough to show that the rank of U(ZG)

coincides with the cardinality of B. For this, first observe that the cardinality

of B is∑d kdtd, where d runs through the divisors of |G|, kd is the number

of cyclic subgroups of G of order d and td is the cardinality of {k : 1 < k <d2 , gcd(d, k) = 1}. Obviously t1 = t2 = 0 and td = φ(d)

2 − 1 for every d > 2.

Therefore, |B| =∑d>2

Äkdφ(d)

2 − kdä

=1+k2+

∑dhd

2 −∑d kd = 1+k2+|G|−2c

2 ,

where hd denotes the number of elements of G of order d (so that h1 = 1 and

h2 = k2) and c is the number of cyclic subgroups of G. By Theorem 1.14.3,

this number coincides with the rank of U(ZG) and the proof is finished.

In Theorem 4.1.5 one can choose, for example, mk,C = φ(|G|), mk,C = φ(|C|)or mk,C = o|C|(k). Observe that the Bass Theorem is the specialization of

108

4.1 abelian groups

Theorem 4.1.5 to G = 〈g〉, a〈gd〉 = gd for d dividing |g|, and mk,〈gd〉 a fixed

multiple of φ(|g|).The advantage of the new proof, with respect to the proofs of Bass and

Bass-Milnor, is that it provides a way to express some power of any given Bass

unit uk,m(g) as a product of a trivial unit and powers of at most 2 elements

from

B =

ßuk,mk,C (aC) : C cyclic subgroup of G, 1 < k <

|C|2, gcd(k, |C|) = 1

™for any given choice of generators aC of cyclic subgroups and integers mk,C as

in Theorem 4.1.5. This is obtained as follows: calculate

• n := |g|; C := 〈g〉 ;• k′1 := the unique integer 0 ≤ k′1 < n such that g = a

k′1C ;

• k′0 := kk′1 mod n;

• for i = 0, 1 : ki := min(k′i, n− k′i); hi :=

®1, if ki = k′i;

ak′iC , otherwise.

• M := lcm (m,mk0,C ,mk1,C) ; c := Mm ;

Then, by equations (3), (6) and (8) from Lemma 1.12.2 we have

uk,m(g)c = uk,M (ak′1C )

= uk′0,M (aC)uk′1,M (aC)−1 = uk0,M (aC)hM0 uk1,M (aC)−1h−M1

= (h0h1−1)Muk0,mk0,C (aC)

Mmk0,C uk1,mk1,C (aC)

− Mmk1,C .

We summarize this result in the following corollary.

Corollary 4.1.6




|C|2, gcd(k, |C|) = 1

™is a virtual basis of U(ZG). Moreover, for any Bass unit uk,m(g) in ZGwe have


for C = 〈g〉, an element h ∈ G and integers c, n0, n1, k0, k1 such that

1 ≤ k0, k1 ≤ |C|2 , g = a±k1C and k0 ≡ ±kk1 mod |C|.

109

central units

4.2 strongly monomial groups

Alternative to Theorem 1.14.2, we give a computation of the rank of the group

of central units in the integral group ring ZG of a finite strongly monomial

group G in terms of its strong Shoda pairs.

Next, we construct generalized Bass units and show that the group they

generate contains a subgroup of finite index in the central units of the integral

group ring ZG for finite strongly monomial groups G. This generalizes a

result of Eric Jespers and Michael M. Parmenter in [JP12, Corollary 2.3] on

generators for central units of the integral group ring of a finite metabelian

group.

Theorem 4.2.1

Let G be a finite strongly monomial group. The rank of Z(U(ZG)) equals

∑(H,K)

Çφ([H : K])

k(H,K)[NG(K) : H]− 1

å,

where (H,K) runs through a complete and non-redundant set of strong

Shoda pairs of G, h is such that H = 〈h,K〉 and

k(H,K) =

ß1 if hhn ∈ K for some n ∈ NG(K);

2 otherwise.

Proof By Proposition 1.7.13, one obtains the following description of the

Wedderburn decomposition of QG:⊕(H,K)

M[G:N ](Q(ζ[H:K]) ∗N/H),

with (H,K) running through a complete and non-redundant set of strong

Shoda pairs of G and N = NG(K).

Since Z(QG) =⊕

(H,K) Q(ζ[H:K])N/H and by Lemmas 1.8.1 and 1.8.2, the

order⊕

(H,K) Z[ζ[H:K]]N/H is the unique maximal order of Z(QG). The rank

of Z(U(ZG)) is the sum of the ranks of the unit groups of Z[ζ[H:K]]N/H , by

Lemma 1.8.6.

Consider the center F = Q(ζ[H:K])N/H of the simple component

M[G:N ](Q(ζ[H:K]) ∗N/H).

110

4.2 strongly monomial groups

Clearly,

[F : Q] =[Q(ζ[H:K]) : Q]

[Q(ζ[H:K]) : F ]=φ([H : K])

[N : H].

Since F is a Galois extension of Q, we know that F is either totally real or

totally complex. If F is totally real, then F is contained in the maximal

real subfield Q(ζ[H:K] + ζ−1[H:K]) of Q(ζ[H:K]). This happens if and only if the

Galois group N/H contains complex conjugation, which means that hhn ∈ Kfor some n ∈ N and h such that H = 〈h,K〉. Now using Dirichlet’s Unit

Theorem 1.8.8, we obtain at once an appropriate rank computation.

Let R be an associative ring with identity. Let x be a unit in R of finite

order n. Let Cn = 〈g〉, a cyclic group of order n. Then the map g 7→ x

induces a ring homomorphism Z 〈g〉 → R. If k and m are positive integers

with km ≡ 1 mod n, then the element

uk,m(x) = (1 + x+ · · ·+ xk−1)m +1− km

n(1 + x+ · · ·+ xn−1)

is a unit in R since it is the image of a Bass unit in Z 〈g〉.In particular, if G is a finite group, M a normal subgroup of G, g ∈ G and

k and m positive integers such that gcd(k, |g|) = 1 and km ≡ 1 mod |g|, then

we have

uk,m(1− M + gM) = 1− M + uk,m(g)M.

Observe that any element b = uk,m(1 − M + gM) is an invertible element of

ZG(1−M)+ZGM . As this is an order in QG, there is a positive integer n such

that bn ∈ U(ZG). Let nG,M denote the minimal positive integer satisfying this

condition for all g ∈ G. Then we call the element

uk,m(1− M + gM)nG,M = uk,mnG,M (1− M + gM)

a generalized Bass unit based on g and M with parameters k and m. Note

that we obtain the classical Bass units of ZG when M = 1.

Before we prove that this construction of generalized Bass units yields a

subgroup of finite index in Z(U(ZG)), we need the following lemma.

Lemma 4.2.2

Let A1 and A2 be finite dimensional Q-algebras such that A1 ⊆ A2 and

consider two orders O1 and O2 in A1 and A2 respectively. Then O2 ∩A1

is an order in A1 and U(O2 ∩A1) and U(O1) are commensurable.

111

central units

Proof Since O2 is a Z-module and A1 is a Q-algebra, clearly O2 ∩ A1 is a

Z-module. Since O2 is a finitely generated Z-module and Z is Noetherian, all

its submodules are finitely generated. Therefore O2 ∩A1 is finitely generated.

Also, since QO2 = A2, it follows that Q(O2 ∩ A1) = A1 and O2 ∩ A1 is an

order in A1. The rest follows from Lemma 1.8.6.

Theorem 4.2.3

Let G be a finite strongly monomial group. The group generated by the

generalized Bass units bnG,H′ with b = uk,m(1 − H ′ + hH ′) for a strong

Shoda pair (H,K) of G and h ∈ H, contains a subgroup of finite index in

Z(U(ZG)).

Proof Let (H,K) be a strong Shoda pair of G. Since H/H ′ is abelian, it

follows from the Bass-Milnor Theorem 1.12.1 that B1(H/H ′) is a subgroup of

finite index in the group of (central) units of Z(H/H ′) ' ZHH ′. A power

of each Bass unit of Z(H/H ′) is the natural image of a Bass unit in ZH.

Hence the group generated by units of the form b = uk,m(1− H ′+ hH ′), with

h ∈ H, is of finite index in U(Z(1− H ′)+ZHH ′). The group generated by the

generalized Bass units bnG,M is still of finite index in U(Z(1 − H ′) + ZHH ′).Let B1 denote this subgroup. Note that B1 is central in ZH.

Since H ′ ⊆ K ⊆ H, we know that ε(H,K) ∈ QHH ′ and hence

Q(1− ε(H,K)) + QHε(H,K) ⊆ Q(1− H ′) + QHH ′.

Since Z(1−ε(H,K))+ZHε(H,K) is an order in Q(1−ε(H,K))+QHε(H,K)

and Z(1 − H ′) + ZHH ′ is an order in Q(1 − H ′) + QHH ′, Lemma 4.2.2

implies that B2 = B1 ∩ (Z(1 − ε(H,K)) + ZHε(H,K)) is of finite index in

U(Z(1− ε(H,K)) + ZHε(H,K)).

If α = 1 − ε(H,K) + βε(H,K) ∈ B2, with β ∈ ZH, and g ∈ NG(K),

then αg = 1 − ε(H,K) + βgε(H,K). Since Z(1− ε(H,K)) + ZHε(H,K) is

a commutative ring, α and αg commute, and thus the product∏g∈NG(K) α

g

is independent of the order of its factors. Since U(ZHε(H,K)) is finitely

generated, it is readily verified that {∏g∈NG(K) u

g : u ∈ U(ZHε(H,K))} is of

finite index in U((ZHε(H,K))NG(K)/H). Hence

B =

∏g∈NG(K)

αg : α ∈ B2 ∩ (1− ε(H,K) + ZHε(H,K))

is a generating set for a subgroup which is of finite index in the unit group

U(Z(1− ε(H,K)) + (ZHε(H,K))NG(K)/H).

112

4.3 abelian-by-supersolvable groups

Furthermore, if T is a transversal of NG(K) in G then

C =

{∏t∈T

γt : γ ∈ B

}

generates a subgroup of finite index in Z(U(Z(1−e(G,H,K))+ZGe(G,H,K))).

To prove that the product in the definition of C is independent on the order

of the product observe that if γ ∈ B and t1, t2 ∈ G then γ = 1 − ε(H,K) +

γ1ε(H,K) for some γ1 ∈ ZH. If t1 6= t2 then ε(H,K)t1ε(H,K)t2 = 0, because

(H,K) is a strong Shoda pair. Using this it is easy to see that γt1 and γt2

commute.

Take now an arbitrary central unit u in Z(U(ZG)). Then we can write this

element as follows

u =∑

(H,K)

ue(G,H,K) =∏

(H,K)

(1− e(G,H,K) + ue(G,H,K)),

where (H,K) runs through a complete and non-redundant set of strong Shoda

pairs of G. Note that conjugates of bnG,H′ = uk,mnG,H′ (1−H ′+hH ′) are again

of this form since conjugates of Bass units are again Bass units and because

(Hg,Kg) is a strong Shoda pair of G if (H,K) is a strong Shoda pair of G for

g ∈ G. Hence the result follows from the previous paragraph.

Note that Theorem 4.2.3 extends the Bass-Milnor Theorem because for

abelian groups the generalized Bass units uk,m(1 − H ′ + hH ′) are precisely

the Bass units.


In Section 4.2, we have proved that the group generated by the generalized

Bass units contains a subgroup of finite index in Z(U(ZG)) for any arbitrary

finite strongly monomial group G. Note that no multiplicatively independent

set for such a subgroup was obtained.

In this section we construct a virtual basis for the center of U(ZG) provided

the finite group G is abelian-by-supersolvable and has the property that every

cyclic subgroup of order not a divisor of 4 or 6 is subnormal in G. The basis

elements are constructed as a (natural) product of conjugates of Bass cyclic

units. For finite nilpotent groups G, constructions of central units of this type

have earlier been considered by Jespers, Parmenter and Sehgal [JPS96]. Ferraz

113

central units

and Simon constructed in [FS08] a basis for the center of U(ZG) in case G is

a metacyclic group of order pq, with p and q two distinct odd prime numbers.

4.3.1 Generalizing the Jespers-Parmenter-Sehgal Theorem

In this section we prove a generalization of a theorem of Jespers, Parmenter

and Sehgal. We say that a subgroup H of a group G is subnormal in G if

there is a finite chain of subgroups of the group, each one normal in the next,

beginning at H and ending at G.

For a finite abelian-by-supersolvable groupG such that every cyclic subgroup

of order not a divisor of 4 or 6, is subnormal in G, the detailed description of

the primitive central idempotents of QG and the Bass-Milnor Theorem 1.12.1

allow us to show that B1(G) contains a subgroup of finite index in Z(U(ZG)).

Furthermore, we obtain a description for the generators of this subgroup.

In order to do this, we first need a new construction for central units based

on Bass units in the integral group ring ZG.

The idea originates from [JPS96], in which the authors constructed central

units in ZG based on Bass units b ∈ ZG for finite nilpotent groups G. One

denotes by Zi the i-th center, i.e. Z0 = 1 and Zi � G is defined such that

Zi/Zi−1 = Z(G/Zi−1). Since G is nilpotent, Zn = G for some n. For any

g ∈ G and a Bass unit b based on g, put b(1) = b, and, for 2 ≤ i ≤ n, put

b(i) =∏h∈Zi

bh(i−1).

By induction, b(i) is independent of the order of the conjugates in the product

expression and b(i) is central in Z 〈Zi, g〉, since for every h ∈ Zi and for every i

there exists x ∈ Zi−1 such that hg = xgh and 〈Zi−1, g〉� 〈Zi, g〉. In particular,

b(n) ∈ Z(U(ZG)).

Note that the previous construction can be modified and improved by con-

sidering the subnormal series 〈g〉 � 〈Z1, g〉 � · · · � 〈Zn, g〉 = G and taking in

each step conjugates in a transversal for Zi in Zi−1. Then, the two construc-

tions differ by a power. The constructions remain valid when starting with an

arbitrary unit u in ZG with support in an abelian subgroup.

We now generalize this construction to a bigger class of groups G. Through-

out G will be a finite abelian-by-supersolvable group such that every cyclic

subgroup of order not a divisor of 4 or 6, is subnormal in G. It is clear that

this class of groups contains the finite nilpotent groups, the dihedral groups

114


D2n =⟨x, y : xn = 1 = y2, yxy = x−1

⟩and the generalized quaternion groups

Q2n =⟨x, y : x2n = 1 = y4, xn = y2, y−1xy = x−1

⟩.

Let u ∈ U(Z 〈g〉), for g ∈ G of order not a divisor of 4 or 6. We consider

a subnormal series N : N0 = 〈g〉 � N1 � N2 � · · · � Nm = G. Now define

cN0 (u) = u and

cNi (u) =∏h∈Ti

cNi−1(u)h,

where Ti is a transversal for Ni in Ni−1. We prove that this construction is

well defined by proving the following three properties.

Lemma 4.3.1

Let g ∈ G, u ∈ U(Z 〈g〉), N , Ni and Ti be as above. We have

(I) cNi−1(u)x ∈ ZNi−1, for all x ∈ Ni;

(II) cNi−1(u)x = cNi−1(u), for all x ∈ Ni−1;

(III) cNi (u) is independent on the choice of transversal Ti.

Proof It is easy to see that equation (II) implies (III). Hence it is sufficient

to prove equations (I) and (II).

We prove these by induction on i. First assume i = 1. Then equations (I)

and (II) are trivial since the support of u is contained in 〈g〉 = N0 �N1.

Now assume the formulas hold for i − 1. Let x ∈ Ni. Then cNi−1(u)x =∏h∈Ti−1

cNi−2(u)hx. By the induction hypothesis we have that cNi−2(u)h ∈ZNi−2 and since Ni−2 � Ni−1 � Ni, also cNi−2(u)hx ∈ ZNi−1, which proves

(I).

Now let x ∈ Ni−1. Then

cNi−1(u)x =∏

h∈Ti−1

cNi−2(u)hx =∏

h′∈Ti−1x

cNi−2(u)h′.

Ti−1x remains a transversal for Ni−1 in Ni−2. Hence, by the induction hypo-

thesis on equation (III), the latter equals cNi−1(u) and we have proved (II).

By equations (I) and (II) we have that the construction is independent of

the order of the conjugates in the product expression. Furthermore, cNm(u),

115

central units

the final step in our construction, is a central unit in ZG, which we will simply

denote by cN (u).

Theorem 4.3.2

Let G be a finite abelian-by-supersolvable group such that every cyclic

subgroup of order not a divisor of 4 or 6, is subnormal in G. Then B1(G)

contains a subgroup of finite index in Z(U(ZG)).

Proof We argue by induction on the order of the group G. For |G| = 1 the

result is clear. So assume now that the result holds for groups of order strictly

less than the order of G.

Because of the Bass-Milnor Theorem 1.12.1, we can assume that G is non-

abelian. Write QG = QG(1− G′)⊕QGG′. It is well known that QG(1− G′)is a direct sum of non-commutative simple rings and QGG′ ' Q(G/G′) is

a commutative group ring. Hence, each z ∈ Z(U(ZG)) can be written as

z = z′ + z′′, with z′ ∈ Z(U(ZG(1 − G′))) and z′′ ∈ U(ZGG′). Note that

z′z′′ = 0 = z′′z′. We will prove that some positive power of z is a product of

Bass units. Since z is an arbitrary element of the finitely generated abelian

group Z(U(ZG)), the result follows.

First we focus on the commutative component. Since G/G′ is abelian, it

follows from the Bass-Milnor Theorem that B1(G/G′) is a subgroup of finite

index in U(Z(G/G′)). A power of each Bass unit of Z(G/G′) is the natural

image of a Bass unit of ZG. Hence, we get that z′′m

=∏ri=1 bi for some

positive integer m and some Bass units bi in ZG, where we denote the natural

image of x ∈ ZG in Z(G/G′) by x. By Proposition 1.12.3, we know that

uk,m(g) has finite order if and only if k ≡ ±1 mod |g|. In particular, there is

a Bass unit based on g ∈ G of infinite order if and only if the order of g is not

a divisor of 4 or 6. Hence we can assume that each bi is based on an element

of order not a divisor of 4 or 6.

By the assumptions on G, we can construct central units in ZG which project

to some power of a bi in Z(G/G′). Indeed, each cNi(bi) is central in ZG,

where Ni is a subnormal series from 〈gi〉 to G when bi is based on gi. Since

Z(G/G′) is commutative, the natural image of cNi(bi) is a power of bi, say

bimi

. Hence z′′m·lcm(mi:1≤i≤r)

=∏ri=1 c

Ni(bi)lcm(mi:1≤i≤r)/mi

. Hence one may

assume there exists some positive integer m′ such that z′′m′

=∏sj=1 c

Nj (bj),

where bj runs through a set of Bass units of ZG with possible repetition.

Therefore, zm′(∏sj=1 c

Nj (bj))−1 = z′′′ + G′, with z′′′ ∈ Z(U(ZG(1− G′))).

116


Since G is abelian-by-supersolvable and hence also strongly monomial, we

know that QG(1− G′) =⊕

(H,K) QGe(G,H,K), where (H,K) runs through a

complete and non-redundant set of strong Shoda pairs of G with QGe(G,H,K)

not commutative. Note that in particular H 6= G for each such strong Shoda

pair.

Let (H,K) be a strong Shoda pair of G with H 6= G. Then it is also

a strong Shoda pair of H and ε(H,K) is a primitive central idempotent

of QH. Since |H| < |G|, the induction hypothesis yields that there ex-

ists a subgroup A1 in B1(H) such that A1 is of finite index in Z(U(ZH)).

Clearly, ZH ⊆⊕

e ZHe, where e runs through all primitive central idempo-

tents of QH. As both ZH and⊕

e ZHe are Z-orders in QH, we have that

Z(U(ZH)) is of finite index in Z(U(⊕

e ZHe)). Hence, A1 is of finite index in

Z(U(⊕

e ZHe)). Since Z(1 − ε(H,K)) + ZHε(H,K) ⊆ Z(⊕

e ZHe), we get

that A = A(H,K) = A1 ∩ (Z(1− ε(H,K)) + ZHε(H,K)) is of finite index in

U(Z(1− ε(H,K))⊕ ZHε(H,K)), and each element of A is a product of Bass

units of ZH.

From Proposition 1.7.13 we know that

QGe(G,H,K) 'M[G:NG(K)](QHε(H,K) ∗ (NG(K)/H))

and its center is isomorphic to (QHε(H,K))NG(K)/H , the fixed subfield of

QHε(H,K) under the action of NG(K)/H. Since U(ZHε(H,K)) is a finitely

generated abelian group, it is easy to verify that ∏n∈NG(K)

un : u ∈ U(ZHε(H,K))

is of finite index in U((ZHε(H,K))NG(K)/H).

Next note that if α = 1 − ε(H,K) + βε(H,K) ∈ A, with β ∈ ZH, then

αn = 1 − ε(H,K) + βnε(H,K), for n ∈ NG(K). Hence, α and αn commute

and thus the product∏n∈NG(K) α

n is independent of the order of its factors.

It follows from the previous that

B = B(H,K) =

∏n∈NG(K)

αn : α ∈ A ∩ (1− ε(H,K) + ZHε(H,K))

is a subgroup of finite index of U(Z(1−ε(H,K))+(ZHε(H,K))NG(K)/H) and

the elements of B are contained in B1(H).

117

central units

Let γ = 1 − ε(H,K) + δ ∈ B, with δ ∈ (ZHε(H,K))NG(K)/H . Let T be

a right transversal of NG(K) in G. Since ε(H,K)tε(H,K)t′

= 0 for different

t, t′ ∈ T , we get that γt and γt′

commute and∏t∈T γ

t = 1 − e(G,H,K) +∑t∈T δ

t ∈ 1− e(G,H,K) + ZGe(G,H,K). Clearly,∏t∈T γ

t corresponds to a

central matrix in QGe(G,H,K) with diagonal entry in (ZHε(H,K))NG(K)/H .

From the previous it follows that C = C(H,K) = {∏t∈T γ

t : γ ∈ B} is

a subgroup of finite index in Z (U(Z(1− e(G,H,K)) + ZGe(G,H,K))). As

each γ ∈ B is an element of B1(H), so is∏t∈T γ

t an element in B1(G).

We can now finish the proof as follows. Write the central unit

z′′′ + G′ =∑

(H,K)

z′′′e(G,H,K) + G′ =∏

(H,K)

(1− e(G,H,K) + z′′′e(G,H,K)),


pairs of G so that QGe(G,H,K) is not commutative and∑

(H,K) e(G,H,K) =

1− G′. Because of the construction of C(H,K), there exists a positive integer

m′′ so that (1 − e(G,H,K) + z′′′e(G,H,K))m′′ ∈ C(H,K) for each (H,K).

Hence (z′′′ + G′)m′′, and thus also

zm′m′′ =

(s∏j=1

cNj (bj)

)m′′(z′′′ + G′)m

′′,

is an element in B1(G).

Remark 4.3.3 Only one argument in the proof of Theorem 4.3.2 makes use

of the assumption that every cyclic subgroup of order not a divisor of 4 or

6, is subnormal in G. It is needed to produce a central unit as a product of

conjugates of a Bass unit b. For that we use the construction cN (b). It is

not clear to us whether an alternative construction exists for other classes of

groups, even for metacyclic groups this is unknown.

At first sight, it appears that one does not use the properties of abelian-

by-supersolvable groups, except for the fact that these groups are strongly

monomial and hence one knows an explicit description of the Wedderburn

components. However, we cannot generalize the proof to strongly monomial

groups since we use an induction hypothesis on subgroups and, unlike the class

of abelian-by-supersolvable groups, the class of strongly monomial groups is

not closed under subgroups.

118


Corollary 4.3.4


subgroup of order not a divisor of 4 or 6, is subnormal in G. For each

such cyclic subgroup 〈g〉, fix a subnormal series Ng from 〈g〉 to G. The

group ⟨cNg (bg) : bg a Bass unit based on g, g ∈ G

⟩is of finite index in Z(U(ZG)).

Proof Because Z(U(ZG)) is finitely generated by Theorem 1.8.7 and using

Theorem 4.3.2, it is sufficient to show that if u = b1b2 · · · bm ∈ Z(U(ZG)),

with each bi a Bass unit based on gi ∈ G, then there exists a positive integer

l so that ul is a product of cNg (bg)’s, with bg a Bass unit based on g ∈ G.

In order to prove this, for each primitive central idempotent e of QG, write

QGe = Mne(De), with ne a positive integer and De a division algebra. If Oeis an order in De, then we have that U(ZG) ∩

∏e GLne(Oe) is of finite index

in U(ZG) (cf. Lemma 1.8.6). It is easy to verify that the central matrices in

SLne(Oe) have finite order.

Now, let u = b1b2 · · · bm ∈ Z(U(ZG)), with each bi a Bass unit based on gi ∈G. Then there exists a positive integer m′ such that um

′,(∏m

i=1 cNgi (bi)

)m′ ∈∏e GLne(Oe). Let ki be the positive integer so that each cNgi (bi) is a product

of ki conjugates of bi. Then cNgi (bi)e and bikie have the same reduced norm.

Hence,

ukm′m∏i=1

cNgi (bi)−m′k/kie ∈ SLne(Oe) ∩ Z(GLne(Oe)),

for k = lcm(ki : 1 ≤ i ≤ m) and thus ukm′∏m

i=1 cNgi (bi)

−m′k/kie is an element

of finite order in Z(GLne(Oe)). Consequently,(ukm

′m∏i=1

cNgi (bi)−m′k/ki

)m′′= 1

for some positive integer m′′, i.e.

ukm′m′′ ∈

⟨cNg (bg) : bg a Bass unit based on g, g ∈ G

⟩.

For finite nilpotent groups of class n, we can always take the subnormal

series Ng : 〈g〉 � 〈Z1, g〉 � · · · � 〈Zn, g〉 = G. Since both constructions cNg (b)

and b(n) only differ by a power, we can deduce the Jespers-Parmenter-Sehgal

result.

119

central units

Corollary 4.3.5 (Jespers-Parmenter-Sehgal)

Let G be a finite nilpotent group of class n. The group⟨b(n) : b ∈ B1(G)

⟩is of finite index in Z(U(ZG)).

4.3.2 Reducing to a basis of products of Bass units

In this section, we reduce the generating set of a subgroup of finite index in

Z(U(ZG)) from Corollary 4.3.4 to a virtual basis of Z(U(ZG)), for G a finite

abelian-by-supersolvable group G such that every cyclic subgroup of order not

a divisor of 4 or 6, is subnormal in G.

First, we need some properties of our construction of central units.

Lemma 4.3.6

Let G be a finite group. Let u, v be units in Z 〈g〉 for g ∈ G and let Nbe a subnormal series N0 = 〈g〉 �N1 � · · · �Nm = G. Assume that h is

a group element of G and denote by N h the h-conjugate of the series N ,

i.e. N h : Nh0 =

⟨gh⟩�Nh

1 � · · ·�Nhm = G. Then

(A) cN (uv) = cN (u)cN (v); and

(B) cNh

(uh) = cN (u).

Proof Let u, v ∈ Z 〈g〉. Then clearly cN0 (uv) = uv = cN0 (u)cN0 (v). By an

induction argument on i, we now get that

cNi (uv) =∏x∈Ti

cNi−1(uv)x =∏x∈Ti

cNi−1(u)xcNi−1(v)x = cNi (u)cNi (v),

for i ≥ 1, since cNi−1(u)x and cNi−1(v)x commute by properties (I) and (II). This

proves equation (A).

Let u ∈ Z 〈g〉 and h ∈ G. We prove that cNh

i (uh) = cNi (u)h by induction

on i. For i = 0, we have cNh

0 (uh) = uh = cN0 (u)h. Let i ≥ 1, now by the

induction hypothesis

cNh

i (uh) =∏x∈Th

i

cNh

i−1(uh)x =∏x∈Th

i

cNi−1(u)hx =∏y∈Ti

cNi−1(u)yh = cNi (u)h.

120


Let G be a group and g ∈ G. Define

Sg = {l ∈ U(Z/|g|Z) : g is conjugate with gl in G}.

In other words, Sg is the image of the homomorphism

NG(〈g〉)→ U(Z/|g|Z) : h 7→ lh,

where lh is the unique element of U(Z/|g|Z) such that gh = glh . The kernel

of this homomorphism is CenG(g). We denote Sg = 〈Sg,−1〉 and we always

assume that transversals of Sg in U(Z/|g|Z) contain the identity 1.

Now we give a generalization of Theorem 4.1.5. For this, we need that a finite

group G, satisfying the assumptions of Theorem 4.3.2, is almost nilpotent.

Lemma 4.3.7

Let Y = {g ∈ G : there exists a subnormal series Ng from 〈g〉 to G}.Then 〈Y 〉 is a nilpotent group.

Proof To prove that 〈Y 〉 is nilpotent, it is sufficient to prove that the p-

elements of 〈Y 〉 form a subgroup, i.e. that products of such elements are again

p-elements. For this, it is sufficient to show that all generators of 〈Y 〉 of order

a power of p commute with all p′-elements in 〈Y 〉.Indeed, assume that all p-elements of 〈Y 〉 commute with all p′-elements.

Let y1, . . . , yk be p-elements, with p prime. Write y1 · · · yk = zpzp′ , with zpa p-element and zp′ a p′-element. Then, by the assumption, zp′ is central in

〈y1, · · · , yk〉. Since the latter group is solvable, we know from the Hall result

that the group 〈y1, · · · , yk〉 contains a subgroup P such that its index is not

divisible by p and a subgroup P ′ such that the only prime dividing its index

is p. From the assumptions, we know that P ′ is central in this group and thus

it follows that 〈y1, · · · , yk〉 = P ×P ′. Hence zp′ ∈ P ′ and y1, . . . , yk ∈ P . This

implies zp′ = 1 and thus y1 · · · yk is a p-element, as desired.

We prove the result by induction on the order of G. If |G| = 1 then the

result is obvious. Note that subgroups of G satisfy the same assumption. So

we assume that the result holds for groups of order less than |G| and also

G = 〈Y 〉.We need to prove that all generators of G = 〈Y 〉 that are p-elements com-

mute with all p′-elements of G. Let x be such a p-element and y such a

q-element. If 〈x, y〉 is a proper subgroup of G then the result holds by the

induction hypothesis. So we may assume that G = 〈x, y〉. We need to prove

121

central units

that x and y commute. Since G is abelian-by-supersolvable, there exists a nor-

mal series 1�A = B0 �B1 � · · ·�Bk = G, with A abelian and each Bi+1/Bia cyclic group of prime order. Clearly, Bk/Bk−1 = 〈x, y〉. Hence without

loss of generality we may assume that Bk/Bk−1 = 〈y〉 and thus Bk/Bk−1 is a

cyclic group of order q. Obviously, Bk−1 is a proper subgroup and thus also

H =¨x, xy

i

, yq | i ∈ N∂⊆ Bk−1 is a proper subgroup of G and thus by the

induction hypothesis it is nilpotent. So, H is a direct product of its Sylow

subgroups. One of its Sylow subgroups is a Sylow p-subgroup, say P , with

x ∈ P . Since this is an invariant subgroup of G, we get that 〈P, y〉 = P o 〈y〉.If this is a direct product then x ∈ P and y commute, as desired.

So suppose that x and y do not commute. Then consider a series

〈y〉�N1 � · · ·�Nl = P o 〈y〉 .

Let i be the smallest index such that Ni contains an element, say z, that does

not commute with y. We may assume that z ∈ P and thus z is a p-element.

Let 1 6= t = z−1y−1zy ∈ Ni−1. In particular, t and y commute. Then zy = zt

and thus z = zyq

= ztq. Now tq = 1. As t is a p-element this implies t = 1, a

contradiction.

Note that necessarily 〈Y 〉 is a characteristic subgroup of G. Hence, by

Lemma 4.3.7, for each g ∈ Y , there exists a series

〈g〉� 〈Z1, g〉� 〈Z2, g〉� · · ·� 〈Zm, g〉 = 〈Y 〉�G,

with Zi the i-th center of 〈Y 〉. Since each 〈Zi, g〉 is normalized by NG(〈g〉), we

can assume that for each g ∈ Y , there exists a subnormal series Ng from 〈g〉to G, which is normalized by NG(〈g〉).

Theorem 4.3.8


subgroup of order not a divisor of 4 or 6, is subnormal in G. Let R denote

a set of representatives of Q-classes of G. For g ∈ R, choose a transversal

Tg of Sg in U(Z/|g|Z) containing 1 and for every k ∈ Tg \ {1}, choose an

integer mk,g with kmk,g ≡ 1 mod |g|. For every g ∈ R, of order not a

divisor of 4 or 6, choose a subnormal series Ng from 〈g〉 to G, which is

normalized by NG(〈g〉). The set{cNg (uk,mk,g (g)) : g ∈ R, k ∈ Tg \ {1}

}is a virtual basis of Z(U(ZG)).

122


Proof For every g ∈ R of order not a divisor of 4 or 6, we choose a subnormal

series Ng from 〈g〉 to G, which is normalized by NG(〈g〉). For each h ∈ G of

order not a divisor of 4 or 6, we agree to choose the subnormal series Nh to

be the x-conjugate of Ng when h = x−1gix, with g ∈ R and i coprime to the

order of g.

By Corollary 4.3.4, the set

B1 = {cNh(uk,m(h)) : h ∈ G, k,m ∈ N, km ≡ 1 mod |h|}

generates a subgroup of finite index in Z(U(ZG)). Let t = φ(|G|). We first

prove that

B2 ={cNg (uk,t(g)) : g ∈ R, k ∈ Tg \ {1}

}generates a subgroup of finite index in Z(U(ZG)). To do so we sieve gradually

the list of units in B1, keeping the property that the remaining units still

generate a subgroup of finite index in Z(U(ZG)), until the remaining units are

the elements of B2. For this, we use the equations from Lemma 1.12.2.

By equation (1), to generate B1 it is enough to use the Bass units of the

form uk,m(h) with h ∈ G, 1 ≤ k < |h| and km ≡ 1 mod |h|. Hence one can

assume that k ∈ U(Z/|h|Z).

By equation (6), for every Bass unit uk,m(h) we have uk,m(h)i = uk,t(h)j

for some positive integers i and j. Thus, by equation (A), units of the

form cNh(uk,t(h)) with k ∈ U(Z/|h|Z) generate a subgroup of finite index

in Z(U(ZG)).

By the definition of a Q-class, we know that each h ∈ G is conjugate to

some gi, for g ∈ R and gcd(i, |g|) = 1. Hence, by equations (3), (A) and (B),

we can reduce further the list of generators by taking only Bass units based

on elements of R.

By equation (4), we can exclude k = 1 and still generate a subgroup of finite

index in Z(U(ZG)).

Let g ∈ G be of order n. We claim that if l ∈ Sg and k ∈ U(Z/nZ) then

cNg (ul,t(gk)) has finite order. As un−l,t(g

k) = ul,t(gk)g−lkt, by (8), we may

assume without loss of generality that l ∈ Sg. By equations (3), (A) and (B)

we have

cNg (uli+1,t(gk)) = cNg (ul,t(g

k))cNg (uli,t(gkl)) = cNg (ul,t(g

k))cNg (uli,t(gk)).

Then, arguing inductively we deduce that

cNg (ul,t(gk))i = cNg (uli,t(g

k)),

123

central units

and in particular cNg (ul,t(gk))t = cNg (ult,t(g

k)) = cNg (u1,t(gk)) = 1, by equa-

tions (1) and (4). This proves the claim.

With g and n as above, every element of U(Z/nZ) is of the form kl with

k ∈ Tg and l ∈ Sg. Using (3) again we have ukl,t(g) = uk,t(g)ul,t(gk). By the

previous paragraph, cNg (ul,t(gk)) has finite order. Hence we can reduce the

generating system and take only k ∈ Tg \ {1}.The remaining units are exactly the elements of B2. Thus 〈B2〉 has finite

index in Z(U(ZG)), as desired.

Let B = {cNg (uk,mk,C (g)) : g ∈ R, k ∈ Tg \ {1}}. Using (6) once more, we

deduce that 〈B〉 has finite index in Z(U(ZG)), since 〈B2〉 does.

To finish the proof we need to prove that the elements of B are mul-

tiplicatively independent. To do so, it is enough to show that the rank

of Z(U(ZG)) coincides with the cardinality of B. It is easy to see that

|B| =(∑

g∈R|Tg|)− |R| and |R| equals the number of Q-classes. By con-

struction, [U(Z/|g|Z) : Sg] equals the number of conjugacy classes contained

in the Q-class of g. Furthermore, [Sg : Sg] = 1 when g is conjugated to

g−1 and [Sg : Sg] = 2 when g is not conjugated to g−1. Therefore |Tg| =

[U(Z/|g|Z) : Sg] is exactly the number of R-classes contained in the Q-class of

g. Hence |B| equals the number of R-classes minus the number of Q-classes

in G. By Theorems 1.14.1 and 1.14.2, this number coincides with the rank of

Z(U(ZG)) and the proof is finished.

4.4 another class within the strongly monomial groups

In Section 4.3, we obtained an explicit description of a virtual basis of the

central units Z(U(ZG)) when G is a finite abelian-by-supersolvable group such

that every cyclic subgroup of order not a divisor of 4 or 6 is subnormal in

G. Note that the latter does not apply to all finite split metacyclic groups

Cm ok Cn, for example if n is a prime number and Cm ok Cn is not abelian

then Cn is not subnormal in Cmok Cn. On the other hand, Ferraz and Simon

did construct in [FS08] a virtual basis of Z(U(Z(Cq o1 Cp))) for p and q odd

and different prime numbers. We extend these results on the construction

of a virtual basis of Z(U(ZG)) to a class of finite strongly monomial groups

containing the metacyclic groups G = Cqmo1Cpn , with p and q different prime

numbers.

We focus on strongly monomial groups G such that there is a complete and

non-redundant set of strong Shoda pairs (H,K) of G with the property that

124


[H : K] is a prime power. For this class of strongly monomial groups, we

construct a virtual basis for the group Z(U(ZG)). The construction of units

in the basis is inspired by the construction of Bass units.

Since the Wedderburn decomposition of QG is well described, we want to

exploit this knowledge to describe a virtual basis of Z(U(ZG)) in terms of

cyclotomic units of QG. In order to compute a virtual basis of Z(U(ZG)), we

will “cover” the central integral units in the different simple components by

using generalized Bass units. This will lead to a final description of the central

units up to finite index. Indeed, take an arbitrary central unit u in Z(U(ZG)).

Then we can write this element as follows

u =∑

(H,K)

ue(G,H,K) =∏

(H,K)

(1− e(G,H,K) + ue(G,H,K)),


pairs of G. Hence it is necessary and sufficient to construct a set of mul-

tiplicatively independent units in the center of each order ZGe(G,H,K) +

Z(1− e(G,H,K)).

The centers of ZGe(G,H,K) + Z(1 − e(G,H,K)) and Z[ζ[H:K]]NG(K)/H +

Z(1−e(G,H,K)) are both orders in the center of the algebra QGe(G,H,K)+

Q(1− e(G,H,K)) and therefore their unit groups are commensurable. Hence,

we are interested in the units of Z[ζ[H:K]]NG(K)/H and furthermore in the

units of ZG projecting to units in Z[ζ[H:K]]NG(K)/H and trivially to the other

components.

We define σr ∈ Aut(〈ζpn〉) ' Gal(Q(ζpn)/Q) by σr(ζpn) = ζrpn . For a

subgroup A of Aut(〈ζpn〉) and u ∈ Q(ζpn), we define πA(u) to be∏σ∈A σ(u).

Since, by assumption, [H : K] equals a prime power, say pn, it is well known

that Aut(H/K) is cyclic, unless p = 2 and n ≥ 3 in which case Aut(H/K) =

〈σ5〉 × 〈σ−1〉. We make abuse of notation to identify NG(K)/H to the Galois

group of Q(ζpn)/Q(ζpn)NG(K)/H and with a subgroup of U(Z/[H : K]Z). The

Galois group Gal(Q(ζpn)/Q(ζpn)NG(K)/H) is a subgroup of Aut(H/K) and it

follows that NG(K)/H is either 〈σr〉 for some r or 〈σr〉× 〈σ−1〉 for some r ≡ 1

mod 4. We simply denote πNG(K)/H by π and have

π(u) =∏

σ∈NG(K)/H

σ(u) =∏

i∈NG(K)/H

ui

for u ∈ Q(ζpn). We will need the following lemma.

125

central units

Lemma 4.4.1

Let A be a subgroup of Aut(〈ζpn〉). Let I be a set of coset representatives

of U(Z/pnZ) modulo 〈A, σ−1〉 containing 1. The set

{πA (ηk(ζpn)) : k ∈ I \ {1}}

is a virtual basis of U(Z[ζpn ]A

).

Proof Assume A = 〈σr〉 or A = 〈σr〉×〈σ−1〉. In both cases, we set l = | 〈σr〉 |.The arguments at the end of the proof of Theorem 4.2.1 show that the unit

group U(Z[ζpn ]A

)has rank pn−1(p−1)

ld − 1 = |I| − 1, where d = 2 if −1 6∈ 〈r〉and d = 1 otherwise. Moreover ld = | 〈A, σ−1〉 |.

By Theorem 1.11.2, the cyclotomic units of the form ηk(ζpn) generate a

subgroup of finite index in U(Z[ζpn ]). Therefore, for every unit u of Z[ζpn ]A,

um =∏hi=1 ηki(ζpn) for some integers m, k1, . . . , kh. Then, um|A| = πA(u) =∏h

i=1 πA(ηki(ζpn)). Hence, it is clear that

{πA(ηk(ζpn)) : k ∈ U(Z/pnZ)}

generates a subgroup of finite index in U(Z[ζpn ]A

).

Throughout the proof we use the equalities from Lemma 1.11.1.

First consider the case when d = 1 (i.e. −1 ∈ 〈r〉). Because of equation (2),

we have

ηrti(ζpn) = ηi(ζpn)ηrt(ζipn),

for i ∈ I and 0 ≤ t ≤ l − 1. Note that πA(ηrt(ζipn)) = πA(ηrt(ζ

irt

pn )), for

0 ≤ t ≤ l − 1. Again by (2), we deduce that

(πA(ηrt(ζipn)))2 = πA(ηrt(ζ

ipn))πA(ηrt(ζ

irt

pn )) = πA(ηr2t(ζipn)),

and hence it follows by induction and equations (1) and (3) that πA(ηrt(ζipn))

has finite order. Hence

{πA(ηi(ζpn)) : i ∈ I}

still generates a subgroup of finite index in U(Z[ζpn ]A

).

Now consider the case when d = 2 (i.e. −1 6∈ 〈r〉). Let J = I∪−I. Then J is

a set of coset representatives of U(Z/pnZ) modulo 〈r〉. By the same arguments

as above, we can deduce that

{πA(ηk(ζpn)) : k ∈ J}

126


generates a subgroup of finite index in U(Z[ζpn ]A

). If i ∈ I, then, by equation

(4), we have that πA(η−i(ζpn)) = πA(−ζ−ipn )πA(ηi(ζpn)) and πA(−ζ−ipn ) is of

finite order. Thus

{πA(ηi(ζpn))} : i ∈ I}

still generates a subgroup of finite index in U(Z[ζpn ]A

).

Now, in both cases, by equation (3), we can exclude i = 1. And since the

size now coincides with the rank, the set

{πA(ηk(ζpn)) : k ∈ I \ {1}}

is a virtual basis of U(Z[ζpn ]A

).

Because of the natural isomorphism Q(H/K) ' QH“K, the following lemma

is a translation of Lemma 4.1.2. It gives a direct link between generalized Bass

units and cyclotomic units.

Lemma 4.4.2

Let H be a finite group, K a subgroup of H and g ∈ H such that H/K =

〈gK〉. Put H = {L ≤ H : K ≤ L}. For every L ∈ H, fix a linear

representation ρL of H with kernel L. Assume that L ∈ H and M is an

arbitrary subgroup of H. Let l = |L∩M |, t = [M : L∩M ] and let k and

m be positive integers such that gcd(k, t) = 1 and km ≡ 1 mod |gu| for

every u ∈M . Then∏u∈M

ρL(uk,mnH,K (gu“K + 1− “K)) = ηk(ρL(g)t)lmnH,K .

Let H be a finite group and K a subgroup of H such that H/K = 〈gK〉 is

a cyclic group of order pn. It follows that the subgroups of H/K form a chain,

hence H = {L : K ≤ L ≤ H} = {Hj =¨gpn−j

,K∂

: 0 ≤ j ≤ n}. This is a

crucial property to prove the next result.

Let k be a positive integer coprime with p and let r be an arbitrary integer.

For every 0 ≤ j ≤ s ≤ n, we construct recursively the following products of

generalized Bass units of ZH:

css(H,K, k, r) = 1,

127

central units

and, for 0 ≤ j ≤ s− 1,

csj(H,K, k, r) =

Ñ∏h∈Hj


h“K + 1− “K)

éps−j−1

Ñs−1∏l=j+1

csl (H,K, k, r)−1

é(j−1∏l=0


).

Here, we agree that by definition an empty product equals 1.

The idea originates from [JdRVG14, Proposition 4.2] for finite elementary

p-groups, where the result arises naturally.

Proposition 4.4.3

Let H be a finite group and K a subgroup of H such that H/K = 〈gK〉is cyclic of order pn. Let H = {Hj =

¨gpn−j

,K∂

: 0 ≤ j ≤ n}. Let k be a

positive integer coprime with p and let r be an arbitrary integer.

Then

ρHj1 (csj(H,K, k, r)) =

®ηk(ζrps−j )

opn (k)ps−1nH,K if j = j1;

1 if j 6= j1,(15)

for every 0 ≤ j, j1 ≤ s ≤ n.

Proof We use a double induction, first on s and, for a fixed s, on s− j. The

minimal cases (s = 1 and s = j) are obvious, so assume that s > 1, j < s and

the result holds for s1 < s and 0 ≤ j, j1 ≤ s1, and for s fixed and j1 > j.

Let u(k) =∏h∈Hj uk,opn (k)nH,K (grp

n−sh“K + 1 − “K). By Lemma 4.4.2, if

0 ≤ i ≤ j, then

ρHi(u(k)) = ηk(ρHi(grpn−s)p

j−i)opn (k)nH,Kp

i

= ηk(ζrpj−i

ps−i )opn (k)nH,Kpi

, (16)

and if j ≤ i ≤ n then

ρHi(u(k)) = ηk(ρHi(grpn−s))opn (k)nH,Kp

j

= ηk(ζrps−i)opn (k)nH,Kp

j

. (17)

By (16) and the induction hypothesis on s, we have

ρHj1 (cs+i−ji (H,K, k, r))

=

®ηk(ζrps−j )

opn (k)nH,Kps+i−j−1

= ρHj1 (u(k))ps−j−1

if j1 = i < j;

1 if j1 6= i < j.(18)

128


By (17) and the induction hypothesis on s− j, we have

ρHj1 (csi (H,K, k, r))

=

®ηk(ζrps−i)

opn (k)nH,Kps−1

= ρHj1 (u(k))ps−j−1

if j < i = j1;

1 if j < i 6= j1.(19)

Now, combining (16), (18) and (19), we have

ρHj1 (csj(H,K, k, r))

=

ρHj (u(k))p

s−j−1

= ηk(ζrps−j )opn (k)nH,Kp

s−1

if j = j1;

ρHj1 (u(k))ps−j−1

ρHj1 (csj1(H,K, k, r))−1 = 1 if j < j1;

ρHj1 (u(k))ps−j−1

ρHj1 (cs+j1−jj1(H,K, k, r))−1 = 1 if j > j1,

as desired.

We state now our main theorem on central units of this section. Observe

that we are again identifying NG(K)/H with a subgroup of U(Z/[H : K]Z)

for a strong Shoda pair (H,K) of G.

Theorem 4.4.4

Let G be a finite strongly monomial group such that there exists a com-

plete and non-redundant set S of strong Shoda pairs (H,K) of G, with

the property that each [H : K] is a prime power. For every (H,K) ∈ S,

let TK be a right transversal of NG(K) in G, let I(H,K) be a set of rep-

resentatives of U(Z/[H : K]Z) modulo 〈NG(K)/H,−1〉 containing 1 and

let [H : K] = pn(H,K)

(H,K) , with p(H,K) a prime number. The set ∏t∈TK

∏x∈NG(K)/H

cn(H,K)

0 (H,K, k, x)t : (H,K) ∈ S, k ∈ I(H,K) \ {1}

is a virtual basis of Z(U(ZG)).

Proof Fix (H,K) ∈ S, N = NG(K), T = TK , I = I(H,K) and n = n(H,K).

It is sufficient to prove that∏t∈T

∏x∈N/H

cn0 (H,K, k, x)t : k ∈ I \ {1}

129

central units

is a virtual basis of the center of 1 − e(G,H,K) + U(ZGe(G,H,K)). To

prove this, we may assume without loss of generality that K is normal in

G, or equivalently N = G. Indeed, assume we can compute a virtual basis

{u1, . . . , us} of the center of 1−ε(H,K)+U(ZNε(H,K)). Each ui is of the form

1−ε(H,K)+viε(H,K) for some vi ∈ ZN and ui−1 = 1−ε(H,K)+v′iε(H,K)

for some v′i ∈ ZN . Then, wi =∏t∈T u

ti = 1−e(G,H,K)+

∑t∈T v

tiε(H,K)t is

a unit in the center of 1−e(G,H,K)+ZGe(G,H,K) since the ε(H,K)t are mu-

tually orthogonal idempotents and they also are orthogonal to 1− e(G,H,K).

Then w1, . . . , ws are independent because so are u1, . . . , us and they form a

virtual basis of the center of 1− e(G,H,K) + ZGe(G,H,K).

From now on we assume that K is normal in G and [H : K] = pn with p a

prime number. Thus N = G and T = {1}. We have to prove that ∏x∈G/H

cn0 (H,K, k, x) : k ∈ I \ {1}

is a virtual basis of the center of 1− e(G,H,K) + U(ZGe(G,H,K)).

Assume first that H = G. Then QGe(G,G,K) ' QG“K ' Q(ζpn). Con-

sider the algebra QGe(G,G,K)+Q(1−e(G,G,K)) as a subalgebra of QG“K+

Q(1− “K). By Proposition 4.4.3, the elements cn0 (G,K, k, 1) project to a cyclo-

tomic unit ηk(ζpn)opn (k)pn−1nG,K in the simple component Q(ζpn) and trivially

in all other components. Since we know that the set {ηk(ζpn) : k ∈ I \ {1}} is

a virtual basis of U(Z[ζpn ]) (cf. Lemma 4.4.1), it follows that

{cn0 (G,K, k, 1) : k ∈ I \ {1}}

is a virtual basis of 1− e(G,G,K) + U(ZGe(G,G,K)).

Now, assume that H 6= G and consider the non-commutative simple com-

ponent QGe(G,H,K) ' QHε(H,K) ∗ G/H with center (QHε(H,K))G/H 'Q(ζpn)G/H ⊆ QH“K. Consider the commutative algebra (QHε(H,K))G/H +

Q(1 − ε(H,K)) as a subalgebra of QH“K + Q(1 − “K). Since H/K is a cyclic

p-group, G/H = 〈σr〉 or G/H = 〈σr〉 × 〈σ−1〉 for some r. Say | 〈σr〉 | = l.

By Lemma 4.4.1, the set {π (ηk(ζpn)) : k ∈ I \ {1}} is a virtual basis of

U(Z[ζpn ]G/H

).

If G/H is cyclic, by Proposition 4.4.3, the elements

cn0 (H,K, k, 1)cn0 (H,K, k, r) · · · cn0 (H,K, k, rl−1)

130


project to π(ηk(ζpn))opn (k)pn−1nH,K in the component Q(ζpn)G/H and trivially

in all other components of QH. Hence the set

{cn0 (H,K, k, 1)cn0 (H,K, k, r) · · · cn0 (H,K, k, rl−1) : k ∈ I \ {1}}

is a virtual basis of Z(U(ZGe(G,H,K) + Z(1− e(G,H,K)))).

If G/H is not cyclic, then the elements

l−1∏i=0

1∏j=0

cn0 (H,K, k, ri(−1)j)

project to a power of π(ηk(ζpn)) in the component Q(ζpn)G/H and trivially in

all other components of QH. Hence also in this case we find a set{l−1∏i=0

1∏j=0

cn0 (H,K, k, ri(−1)j) : k ∈ I \ {1}

},

which is a virtual basis of Z(U(ZGe(G,H,K) + Z(1− e(G,H,K)))).

We apply our results to construct a virtual basis of the group Z(U(ZG)),

for the metacyclic groups of the form Cqm o1 Cpn , for two different prime

numbers p and q. This generalizes results from Ferraz and Simon where the

case m = n = 1 is handled [FS08].

Corollary 4.4.5

Let p and q be different prime numbers. Let G = Cqm o1 Cpn be a finite

metacyclic group with Cpn = 〈b〉 and Cqm = 〈a〉. Let r be such that

ab = ar. For each j = 1, . . . ,m, let Ij be a set of coset representatives of

U(Z/qjZ) modulo 〈r,−1〉.If p = 2, then

U =¶ci0(G,

¨a, b2

i∂, k, 1) : 1 < k < 2i−1, 2 - k, i = 2, . . . , n

©⋃ ß

cmm−j(〈a〉 , 1, k, 1)cmm−j(〈a〉 , 1, k, r) · · · cmm−j(〈a〉 , 1, k, r2n−1) :

k ∈ Ij \ {1}, j = 1, . . . ,m

™is a virtual basis of Z(U(ZG)), consisting of 2n−1 + qm−1

2n − n−m units.

131

central units

If p is odd, then

U =

ßci0(G,

ä, bp

i∂, k, 1) : 1 < k <

pi

2, p - k, i = 1, . . . , n

™⋃ ß

cmm−j(〈a〉 , 1, k, 1)cmm−j(〈a〉 , 1, k, r) · · · cmm−j(〈a〉 , 1, k, rpn−1) :

k ∈ Ij \ {1}, j = 1, . . . ,m

™is a virtual basis of Z(U(ZG)), consisting of pn−1

2 + qm−12pn − n−m units.

Proof We first compute the rank of Z(U(ZG)) using the formula from Theo-

rem 4.2.1. By Corollary 1.7.18, the strong Shoda pairs of G are of the following

forms:

1.ÄG,Li :=

ä, bp

i∂ä, i = 0, . . . , n;

2.Ä〈a〉 ,Kj :=

äqj∂ä, j = 1, . . . ,m.

When p is odd, an easy computation shows that the rank equals

n∑i=1

Åpi−1(p− 1)

2− 1

ã+

m∑j=1

Åqj−1(q − 1)

2pn− 1

ã=pn − 1

2+qm − 1

2pn− n−m,

because r has odd order modulo qj , j = 1, . . . ,m.

When p = 2, the rank equals

n∑i=2

(2i−2 − 1

)+

m∑j=1

Åqj−1(q − 1)

2n− 1

ã= 2n−1 +

qm − 1

2n− n−m,

since ab2n−1

= a−1, because r has even order modulo qm.

Now we use Theorem 4.4.4 to obtain a virtual basis of Z(U(ZG)):ßci0(G,Li, k, 1) : i = 1, . . . , n, 1 < k <

pi

2, p - k

™⋃{

pn−1∏x=0

cj0(〈a〉 ,Kj , k, rx) : j = 1, . . . ,m, k ∈ Ij \ {1}

},

132

4.5 conclusions

where Ij is a set of coset representatives of U(Z/qjZ) modulo 〈r,−1〉 containing

1. We claim that the units cj0(〈a〉 ,Kj , k, rx), which are products of generalized

Bass units, can be replaced by cmm−j(〈a〉 , 1, k, rx), which are products of Bass

units. Indeed, these units project trivially on the commutative algebra QGa.

Moreover, by Proposition 4.4.3, they project to the unit

ηk(ζqj )oqm (k)qm−1

in the simple component QGε(〈a〉 ,Kj) ' Q(ζqj ) and trivially in all other

components of QG(1− a). By Lemma 4.4.1, the set

{π(ηk(ζqj )

): k ∈ Ij \ {1}}

is a virtual basis of U(Z[ζqj ]

Cpn). This proves the claim.

As we have shown, the conditions in the statement of Theorem 4.4.4 on

the strong Shoda pairs of the group G are fulfilled when G is a metacyclic

group of the type Cqm o1 Cpn , for two different prime numbers p and q. How-

ever, the class of strongly monomial groups such that there is a complete

and non-redundant set of strong Shoda pairs (H,K) of G with the property

that [H : K] is a prime power, is a wider class. For example the alternating

group A4 of degree 4 satisfies the condition and it is not metacyclic. Al-

though, not all strongly monomial groups have only strong Shoda pairs with

prime power index. It can be shown that all strong Shoda pairs of the dihe-

dral group D2n =⟨a, b : an = b2 = 1, ab = a−1

⟩(respectively, the quaternion

group Q4n =⟨x, y : x2n = y4 = 1, xn = y2, xy = x−1

⟩) have prime power in-

dex if and only if n is a power of a prime number (respectively, n is a power

of 2).

4.5 conclusions

We gave a construction for a virtual basis of Z(U(ZG)) within B1(G) in the

case when G is a finite abelian-by-supersolvable group such that every cyclic

subgroup of order not a divisor of 4 or 6, is subnormal in G. For finite strongly

monomial groups G, we have proved that the generalized Bass units contain

a subgroup of finite index in Z(U(ZG)). Moreover, when G is a strongly

monomial group such that there is a complete and non-redundant set of strong

Shoda pairs (H,K) of G with the property that each [H : K] is a prime power,

a virtual basis within the generalized Bass units is constructed. In all results so

133

central units

far, one proves that the (generalized) Bass units generate a subgroup of finite

index in Z(U(ZG)). However, there are no examples known where B1(G) does

not contain a subgroup of finite index in Z(U(ZG)). Therefore, one could ask

and investigate whether this is true in general for all finite groups.

A continuation of this work would stem from new constructions of central

units for other classes of finite groups, both strongly monomial groups and

non-strongly monomial. Another direction would be to generalize the known

results to Z(U(RG)) for the ring of integers R of a number field.

134

5A P P L I C AT I O N S T O U N I T S O F G RO U P R I N G S

In this chapter, we show how the description of a complete set of orthogonal

idempotents in QG together with a description of the units in Z(U(ZG)) leads

to a description of the unit group U(ZG), up to finite index, for finite groups

G which have no exceptional components in the Wedderburn decomposition of

QG. We do this for the metacyclic groups Cqm o1 Cpn . This is merely based

on the results from Theorems 1.9.2 and 1.9.3.

We also show how to overcome the lack of a result as Theorem 1.9.3 in the

case when QG contains an exceptional component of type EC2, but none of

type EC1. If one can not construct central units, one can just allow all Bass

units and use Proposition 1.13.2.

Recall that finite groups G having an exceptional component in QG are clas-

sified in Corollary 3.3.1. This classification distinguishes between exceptional

components of type EC1 and the ones of type EC2.

5.1 a subgroup of finite index in U (Z(Cqm o1 Cpn ))

As an application of Theorem 1.7.19 and Corollary 4.3.5, Eric Jespers, Gabriela

Olteanu and Angel del Rıo obtained a factorization of a subgroup of finite index

of U (ZG) into a product of three nilpotent groups, and they explicitly con-

structed finitely many generators for each of these factors [JOdR12, Theorem

5.3].

Throughout this section p and q are different prime numbers, m and n are

positive integers and G = Cqm o1Cpn . As an application of the description of

the matrix units in each simple component QGe (Corollary 2.2.5) and of the

description of the central units in ZG (Corollary 4.4.5), we construct explicitly

finitely many generators for three nilpotent subgroups of U (ZG) that together

generate a subgroup of finite index.

135

applications to units of group rings

In order to state the next theorem, it is convenient to introduce the notation

of class sum. Let G be a finite group, X a normal subgroup of G and Y a

subgroup of G such that Y acts faithfully on X by conjugation. Consider the

orbit xY of an element x ∈ X , then we will call ›xY =∑

y∈Y xy ∈ ZX the

orbit sum of x. By X Y we will denote the set of all different orbit sums ›xYfor x ∈ X .

Theorem 5.1.1

Let p and q be different prime numbers. Let G = Cqm o1 Cpn be a

finite metacyclic group with Cpn = 〈b〉 and Cqm = 〈a〉. Assume that

either q 6= 3, or pn > 2. For every j = 1, . . . ,m, let Kj =¨aqj∂, let

Fj be the center of QGε(〈a〉 ,Kj), fix a normal element wj of Q(ζqj )/Fjand let ψj : QGε(〈a〉 ,Kj)→Mpn(Fj) be the isomorphism given by The-

orem 1.5.2 with respect to wj . Let xj = ψj−1(P )bε(〈a〉 ,Kj)ψj

−1(P )−1,

with

P =

1 1 1 · · · 1 1

1 −1 0 · · · 0 0

1 0 −1 · · · 0 0...

......

. . ....

...

1 0 0 · · · −1 0

1 0 0 · · · 0 −1

,

and tj be a positive integer such that tjxkj ∈ ZG for all k with 1 ≤ k ≤ pn.

Then the following two groups are finitely generated nilpotent subgroups

of U(ZG):

V +j =

≠1 + pnt2jyx


∑,

V −j =

≠1 + pnt2jyx


∑.

Hence V + =∏mj=1 V

+j and V − =

∏mj=1 V

−j are nilpotent subgroups of

U(ZG). Furthermore, the group⟨U, V +, V −

⟩,

with U as in Corollary 4.4.5, is of finite index in U(ZG).

136

5.2 a method to compute U (ZG) up to commensurability

Proof It is enough to show that for each primitive central idempotent of QG,

the group 〈U, V +, V −〉 contains a subgroup of finite index in U(Z(1−e)+ZGe).Since the units of the commutative components are central, we only have to

consider the non-commutative components QGe ' Mpn

ÅQÅfl〈a〉〈b〉ã eã, with

e = ε (〈a〉 ,Kj), for j = 1, . . . ,m, from Corollary 1.7.18. Let O = Zïfl〈a〉〈b〉ò e,

which is as a Z-module finitely generated byfl〈a〉〈b〉e. Clearly, for y ∈ fl〈a〉〈b〉,

the elements of the form (1 − e) + (e + pnt2jyxhj bx

kj ) are in ZG and project

trivially to QG(1− e). By Corollary 2.2.5, the group 〈V +j , V

−j 〉, generated by

these elements, projects to the group

〈1 + zjEhk : zj ∈ pnt2jO, 1 ≤ h, k ≤ pn, i 6= j, Ehk a matrix unit〉

of elementary matrices of Mpn(O).

If pn > 2, then the conditions of Theorem 1.9.2 are clearly satisfied. If

p = 2, n = 1 and q 6= 3, the conditions of Theorem 1.9.3 are satisfied since

U(O) is finite if and only if j = 1 and q = 3. This can be seen by computing

the rank of U(O) for which a formula is given in Corollary 4.4.5. Hence in

all cases 〈V +j , V

−j 〉 ⊆ U(ZG) is a subgroup of finite index in 1− e+ SLpn(O).

By Corollary 4.4.5, U has finite index in Z(U(ZG)) and therefore it contains

a subgroup of finite index in the center of 1− e+ GLpn(O). Since the center

of GLpn(O) together with SLpn(O) generates a subgroup of finite index in

GLpn(O), it follows that 〈U, V +, V −〉 contains a subgroup of finite index in

the group of units of Z(1 − e) + ZGe. Now the statement follows, since V +j

and V −j correspond to upper and lower triangular matrices.


The proof of the following proposition is based on the Euclidean algorithm.

Proposition 5.2.1

Let O be a left norm Euclidean order with norm N in either Q, a totally

definite quaternion algebra over Q, or a quadratic imaginary extension of

Q. If B is a Z-basis of O, then the set

X =

ßÅ1 x

0 1

ã,

Å1 0

x 1

ã: x ∈ B

™137


generates a subgroup of finite index in SL2 (O).

Proof We first remark that the elementary matrices E2(O) are generated by

X, because of the following matrix computations for n ∈ Z, x, y ∈ O:Å1 nx

0 1

ã=

Å1 x

0 1

ãn,Å

1 x+ y

0 1

ã=

Å1 x

0 1

ãÅ1 y

0 1

ã,

and the transposed equalities.

Take now

Åa b

c d

ã∈ SL2(O). We first consider the case where both a and

c are non-zero. Since O is left norm Euclidean, there exists q1, r1 ∈ O such

that a = q1c+ r1 with N(r1) < N(c). An easy computation shows thatÅ1 −q1

0 1

ãÅa b

c d

ã=

År1 b− q1d

c d

ã.

Now there exists q2, r2 ∈ O such that c = q2r1 + r2 with N(r2) < N(r1) givingÅ1 0

−q2 1

ãÅr1 b− q1d

c d

ã=

År1 b− q1d

r2 −q2b+ q2q1d+ d

ã.

Repeating this argument and since the sets {N(x) < N(a) : x ∈ O} and

{N(x) < N(c) : x ∈ O} are finite, one can assume that there exists a

matrix M ∈ 〈X〉 such that

Åa b

c d

ãis either of the form M

Å∗ ∗0 ∗

ãor

M

Å0 ∗∗ ∗

ã.

Now assume that c = 0. We compute that

Åa b

0 d

ã∈ SL2(O) if and only

if N(a)N(d) = 1 and this happens if and only if a, d ∈ U(O). We can conclude

that Åa b

0 d

ã=

Å1 bd−1

0 1

ãÅa 0

0 d

ã.

When a = 0, then b, c ∈ U(O) andÅ0 b

c d

ã=

Å1 0

db−1 1

ãÅ0 b

c 0

ã.

138


Because of Dirichlet’s Theorem 1.8.8 and Kleinert’s Theorem 1.9.4, U(O) is

finite. Hence, there are only finitely many matrices of the form

Åa 0

0 d

ãandÅ

0 b

c 0

ã. This means that X generates SL2(O) up to finite index.

Due to the restrictions we obtained on the possible exceptional components

in Corollary 1.9.9, together with Proposition 5.2.1, we generalize Proposi-

tion 1.13.3 and Corollary 1.13.4. That is, we allow exceptional components

M2(D) ' QGe of type EC2, provided one can establish a concrete isomor-

phism M2(D) → QGe. The proof we present is an adapted version of the

proofs of [JL93, Theorem 3.3, Corollary 4.1]. For reasons of completeness, we

reuse a part of their proofs.

Proposition 5.2.2

Let G be a finite group and let QG =⊕n

i=1 QGei '⊕n

i=1Mni(Di) be

the Wedderburn decomposition of QG. Assume that QG does not contain

exceptional components of type EC1. Also, assume that for each integer

i ∈ {1, . . . , n} such that ni 6= 1 and QGei is not exceptional (of type EC2),

Gei is not fixed point free.

For every exceptional component QGei ' M2(Di), Di has a left norm

Euclidean order Oi. Take a Z-basis Bi of Oi and let ψi : M2(Di)→ QGeibe a Q-algebra isomorphism. For such i, set

Ui :=

ß1 + ψi

Å0 x

0 0

ã, 1 + ψi

Å0 0

x 0

ã: x ∈ Bi

™.

The subgroup U := 〈B1(G) ∪ B2(G) ∪⋃i Ui〉 of QG is commensurable

with U(ZG).

Proof Let i ∈ {1, . . . , n} and take a maximal order Oi in the division ring

Di. Since the natural image of B1(G) in K1(ZG) is of finite index, it suffices

to verify condition 2 of Proposition 1.13.2 for each simple component QGei,i.e. verify that U ∩ (1− ei + SLni(Oi)) is of finite index in 1− ei + SLni(Oi).

If ni = 1, the result follows trivially since Di is not exceptional and hence

either is commutative or a totally definite quaternion algebra. In both cases

SL1(Oi) is finite (Theorem 1.9.4).

If ni = 2 and M2(D) is exceptional of type EC2, then D equals Q, a

quadratic imaginary extension of Q or a totally definite quaternion algebra

139


over Q. We know by Corollary 1.9.11 that Di has a left norm Euclidean order

which is the unique maximal order. Hence Oi is left norm Euclidean and it

follows from Proposition 5.2.1 that Ui ∩ (1 − ei + SLni(Oi)) is of finite index

in 1− ei + SLni(Oi).For the remaining components QGei, we claim that there always exists a

gi ∈ G such that “giei is a non-central idempotent in QGei. Let e be a prim-

itive central idempotent of CGei and let ρ be the irreducible representation

ρ : Gei → CGe. Since Gei is not fixed point free, there exists a gi ∈ G such

that giei 6= ei and ρ(giei) has 1 as an eigenvalue. Diagonalizing ρ(giei), we

have

ρ(giei) =

1 · · · 0 0 · · · 0...

. . ....

.... . .

...

0 · · · 1 0 · · · 0

0 · · · 0 ζj · · · 0...

. . ....

.... . .

...

0 · · · 0 0 · · · ζni

,

with 2 ≤ j ≤ ni and all ζj , . . . , ζni different from 1. Consequently,

ρ(“giei) =

1 · · · 0 0 · · · 0...

. . ....

.... . .

...

0 · · · 1 0 · · · 0

0 · · · 0 0 · · · 0...

. . ....

.... . .

...

0 · · · 0 0 · · · 0

.

It follows that ρ(“giei) 6= 0 and ρ(“giei) 6= 1. Hence “giei 6= 0, “giei 6= ei and “gieiis a non-central idempotent in QGei.

Consider now the generalized bicyclic units

Bi =⟨1 + z2

i (1− “giei)h“giei, 1 + z2i“gieih(1− “giei) : h ∈ G

⟩,

with zi a minimal positive integer such that zi“giei ∈ ZG. Note that, for k, l ∈ Zand x, y ∈ G,

(1 + z2i (1−“giei)x“giei)k(1 + z2

i (1−“giei)y“giei)l = 1 + z2i (1−“giei)(kx+ ly)“giei.

Hence

{1 + z2i (1− “giei)α“giei, 1 + z2

i“gieiα(1− “giei) : α ∈ ZG} ⊆ Bi.

140


Since “giei is a non-central idempotent in QGei 'Mni(Di), there exist matrix

units Ekl, 1 ≤ k, l ≤ ni such that “giei = E11 + . . .+Emm with 1 ≤ m < ni. If

k ≤ m,m+ 1 ≤ l ≤ ni, then“gieiOiEkl(1− “giei) = OiEkl.

Hence, as Oi is a finitely generated Z-module, there exists a positive integer

nkl such that

1 + nklOiEkl ⊆ Bi.

And similarly,

1 + nlkOiElk ⊆ Bi,

for some positive integer nlk. It follows that there exists a positive integer x

such that 1 + xOiEkl ∈ Bi and 1 + xOiElk ∈ Bi for all 1 ≤ k ≤ m,m + 1 ≤l ≤ ni. Now let 1 ≤ k, l ≤ m, k 6= l and α ∈ Oi. Then

1+x2αEkl = (1−xαEk,m+1)(1−xEm+1,l)(1+xαEk,m+1)(1+xEm+1,l) ∈ Bi.

Similarly, for m + 1 ≤ k, l ≤ ni and k 6= l, it follows that 1 + x2OiEkl ⊆ Bi.

Take now Ii = x2Oi. This is a non-zero ideal of Oi. Then Theorems 1.9.2

and 1.9.3 imply that 1−ei+Eni(Ii) ⊆ Bi is of finite index in 1−ei+SLni(Oi).By Lemma 1.13.1, it follows that Bi ⊆ B2(G) and hence B2(G) contains a

subgroup of finite index in 1− ei + SLni(Oi) and the proposition follows.

The set U from Proposition 5.2.2 is constructive if one can determine an

explicit isomorphism ψi : M2(Di) → QGei, for each exceptional component

QGei. Recall that all exceptional components of type EC2 are of the follow-

ing form: M2(Q), M2(Q(√−1)), M2(Q(

√−2)), M2(Q(

√−3)), M2

Ä−1,−1

Q

ä,

M2

Ä−1,−3

Q

ä, M2

Ä−2,−5

Q

ä; and their maximal orders are described in Corol-

lary 1.9.11. In order to describe the explicit isomorphisms ψi, one needs to

construct non-central idempotents in QGei. Also, for the non-exceptional com-

ponents QGej , we need non-central idempotents and therefore we ask Gej to

be not fixed point free. If however, some Gej is fixed point free, but one does

know another construction of non-central idempotents in QGej , then one can

modify Proposition 5.2.2 and add the generalized bicyclic units based on such

non-central idempotents to the set U . In this way, U still yields a subgroup

commensurable with U(ZG).

Some generic constructions of non-central primitive idempotents are given

for nilpotent groups in Theorem 1.7.19 and for finite strongly monomial groups,

with a trivial twisting for all strong Shoda pairs, in Theorem 2.2.1.

141


5.3 examples

5.3.1 U(ZD+16) up to finite index

Let G be the group with presentation D+16 =

⟨a, b : a8 = 1, b2 = 1, bab = a5

⟩.

This is the group with SmallGroup ID [16,6]. However, this example is

already well studied in [JL91, Corollary 4.10], it nicely demonstrates the use

of our method.

Using wedderga, we compute the Wedderburn decomposition:

QG = 4Q⊕ 2Q(i)⊕M2(Q(i)).

The primitive central idempotent e associated to the last simple component

is afforded by the pair (〈a〉 , 1) and equals e = e(G, 〈a〉 , 1) = 12 −

12a

4. Hence

this simple component equals the crossed product QGe = Q 〈a〉 ε(〈a〉 , 1) ∗ 〈b〉.It is easy to see that eb is a non-trivial idempotent of QGe, which affords

a description of QGe as M2(bQGeb). Another simple calculation shows that

(ba2eb)2 = −eb and hence the map ba2eb →Åi 0

0 0

ãdefines a Q-algebra

isomorphism between QGe and M2(Q(i)).

To determine an explicit isomorphism ψ : M2(Q(i)) → QGe, it suffices to

find images of the following elements:

E :=

Å1 0

0 0

ã, I :=

Åi 0

0 0

ã, A :=

Å0 1

0 0

ã, B :=

Å0 0

1 0

ã.

We already know possible images for E and I:

ψ(E) = eb ∈ 1

4ZG, ψ(I) = ba2eb ∈ 1

4ZG.

The images of A and B must satisfy

ψ(A) · ψ(B) = ψ(E), (20)

ψ(E) · ψ(A) · (1− ψ(E)) = ψ(A) (21)

and

(1− ψ(E)) · ψ(B) · ψ(E) = ψ(B). (22)

142

5.3 examples

Define

ψ(A) = bae(e− b) = bae(1− b) ∈ 1

4ZG and ψ(B) = (1− b)a−1eb ∈ 1

4ZG.

Equations (21) and (22) are satisfied by definition and one verifies that

ψ(A)ψ(B) = eb = ψ(E).

Now we apply Proposition 5.2.2 to compute U(ZG) up to finite index. Since

the components Q and Q(i) yield a finite group of units in any order, it suf-

fices to compute the group of units of 1 − e + ZGe up to finite index, for

QGe 'M2(Q(i)).

Let Z[i] be the maximal order of Q(i), then U(ZGe) and SL2(Z[i]) are com-

mensurable. From Proposition 5.2.2, we conclude that the elements

1 + ψ

Å0 1

0 0

ã= 1 + bae(1− b), 1 + ψ

Å0 0

1 0

ã= 1 + (1− b)a−1eb,

1 + ψ

Å0 i

0 0

ã= 1 + ba2bae(1− b), 1 + ψ

Å0 0

i 0

ã= 1 + (1− b)a−1ba2eb

generate U(1 − e + ZGe) up to commensurability. Hence this set of elements

in QG generates a group which is commensurable with U(ZG).

However these images do not lie in ZG but in 14ZG. If one is not satisfied

with commensurability, but if one wants to construct a list of generators within

ZG, then one has to deduce from the generators of SL2(Z[i]) a list of generators

of the congruence subgroup

C =

ßÅa+ 1 b

c d+ 1

ã: a, d ∈ 2Z[i], b, c ∈ 4Z[i], (a+ 1)(d+ 1)− bc = 1

™using techniques as for example Schreier’s Lemma, as shown in [JL91, Corollary

4.10]. It is easy to verify that {1 + ψ(x− 1) : x ∈ C} is contained in ZG and

is of finite index in U(ZG). Schreier’s Lemma leads to lengthy and technical

computations and therefore we opted to not include them in this thesis. More

details can be found in [EKVG15].

5.3.2 U(ZSL(2, 5)) up to commensurability

By Theorem 1.1.2, we know that SL(2, 5) is the smallest non-solvable Frobe-

nius complement (i.e. fixed point free group). We apply Proposition 5.2.2 to

143


investigate the units of ZSL(2, 5). Since none of the older techniques, such

as Proposition 1.13.3 and Corollary 1.13.4, apply here, this example shows

the strength of our method. Using wedderga, we compute that QSL(2, 5) is

isomorphic to

Q⊕M4(Q)⊕Ç−1,−1

Q(√

5)

å⊕M2

Å−1,−3

Q

ã⊕M5(Q)⊕M3

Å−1,−1

Q

ã⊕M3(Q(

√5)).

An easy GAP computation shows that for all simple components QGe differ-

ent from Q and(−1,−1

Q(√

5)

), there exists a group element g such that g projects to

a non-central idempotent in QGe. Therefore, we can apply Proposition 5.2.2.

Consider the maximal order O = Z[1, i, 1

2 (1 + j), 12 (i+ ij)

]in the quater-

nion algebraÄ−1,−3

Q

ä. Let e be the primitive central idempotent associated to

the component M2

Ä−1,−3

Q

äand let ψ be the isomorphism between M2

Ä−1,−3

Q

äand QGe. Let U be the subset of QG containing the elements

1 + ψ

Å0 1

0 0

ã, 1 + ψ

Å0 0

1 0

ã, 1 + ψ

Å0 i

0 0

ã, 1 + ψ

Å0 0

i 0

ã,

1 + ψ

Å0 1

2 (1 + j)

0 0

ã, 1 + ψ

Å0 0

12 (1 + j) 0

ã,

1 + ψ

Å0 1

2 (i+ ij)

0 0

ã, 1 + ψ

Å0 0

12 (i+ ij) 0

ã.

Then the group 〈U ∪ B1(G) ∪ B2(G)〉 is commensurable with U(ZG).

5.4 conclusions

We have shown how a concrete algebra isomorphism M2(Di) → QGei for all

exceptional components M2(Di) of type EC2 leads to a finite generating set

(up to commensurability) of the group of units of ZG of a finite group G,

provided that G does not contain exceptional components of type EC1 and

all non-exceptional components are not fixed point free. Hence, to apply the

technique to other finite groups than the ones presented, one should start to

study the finite list of groups in Table 2 and find concrete isomorphisms of the

corresponding exceptional simple component M2(D)→ QGe.

144

5.4 conclusions

To extend the techniques, a first thing to do is to find constructions of non-

central idempotents in the rational group algebra of fixed point free groups.

Secondly, one should investigate constructions of units in orders of division

rings. Recently some progress has been made in the latter topic, see [CJLdR04,

JPSF09, JJK+15, BCNS].

Alternatively, the results in Chapter 2 possibly extend to group rings over

number fields F . Let R be the ring of integers of F . If one can construct a gen-

erating set of Z(U(RG)), then one extends the techniques from Theorem 5.1.1

to U(RG) to generate the unit group up to finite index.

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154

I N D E X

2-part, 22

2′-part, 22

F -critical, 72

G-code, 61

Q-class, 39

Q-conjugate, 39

R-class, 39

R-conjugate, 39

Z-order, 24

p-adic valuation, 5

p-local index, 7

q-cyclotomic class, 54

action, 10

archimedean valuation, 4

Bass unit, 34

bicyclic unit, 36

Cauchy sequence, 5

central simple algebra, 4

central unit, 39

classical crossed product, 10

commensurable, 25

complete, 5

complete and non-redundant set of

Shoda pairs, 18

complete and non-redundant set of

strong Shoda pairs, 19

congruence subgroup, 26

Congruence Subgroup Problem, 26

crossed product, 10

CSP, 26

cyclic cyclotomic algebra, 10

cyclotomic unit of QG, 34

cyclotomic unit of Q(ζn), 34

degree, 4

elementary matrix, 27

equivalent valuations, 4

exceptional component, 28

exceptional simple algebra, 28

field of character values, 16

finite prime, 5

fixed point free group, 2

Frobenius complement, 2

Frobenius group, 2

generalized Bass unit, 111

generalized bicyclic unit, 36

group algebra, 12

group code, 61

group ring, 12

idempotent, 14

induced character, 17

infinite prime, 5

left G-code, 61

left group code, 61

left norm Euclidean, 30

length, 61

linear code, 61

local Hasse invariant, 6

local index, 5, 7

local Schur index, 5

155

matrix unit, 14

maximal order, 24

monomial character, 17

monomial group, 17

non-archimedean valuation, 4

normal, 3

normal basis, 3

number field, 4

orbit sum, 136

order, 24

place, 5

prime, 5

prime lying over, 7

primitive central idempotent, 15

primitive idempotent, 14

quaternion algebra, 3

ramification index, 7

rank, 25, 61

reduced norm, 26

residue degree, 7

Schur index, 4

semisimple, 13

Shoda pair, 17

split, 6

strong Shoda pair, 18

strongly monomial character, 18

strongly monomial group, 18

subnormal, 114

support, 12

totally definite, 3

twisting, 10

unit group, 25

valuation, 4

virtual basis, 101

Z-group, 31

156

colophon

The research presented in this PhD thesis was supported by The Research

Foundation - Flanders (FWO) and the Vrije Universiteit Brussel (VUB).

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