SATURATED FORMATIONS, SCHUNCK CLASSES

AND THE STRUCTURE OF FINITE GROUPS

Abdulmotalab M.O. Ihwil

B.Sc., A L - Fateh Univers i ty , T r i p o l i , Libya, 1979

A THESIS SUBb4ITTED I N PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE

i n t h e Department

0 f

Mathematics and S t a t i s t i c s

@ Abdulmotalab M . O . Ihwi l , 1986

SIMON FRASER UNIVERSITY

Apr i l 1986

A l l r i g h t s reserved. This t h e s i s mhy not be reproduced i n whole o r i n p a r t , by photocopy

o r o t h e r means, without permission of t h e au thor .

APPROVAL

Name :

Degree:

T i t l e of Thesis :

Abdulmotalab M.O. Ihwil

Master of Science

Saturated formations, Schunck c l a s s e s and t h e s t r u c t u r e

of f i n i t e groups.

Chairman : . C . Vi l l egas

n Dr, N . R . Reilly

,dl " I /r Ti. A . R . Freedman Externa l Examiner

Assoc ia te Professor Department of Mathematics and S t a t i s t i c s

Simon F rase r Un ive r s i t y

Date approved: A p r i l 3 0 1 lgg6

PARTIAL COPYRIGHT LICENSE

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T i t l e of Thes i s/Project/Extended Essay

Author:

( s igna tu re )

tnlah P f . r ) . T M l

( name

APRIL 30,1986

(da te)

Abstrac t

This i s a s tudy of t h e s t r u c t u r e of f i n i t e groups from t h e s tandpoint

of c e r t a i n c l a s s e s of groups. Examples of such c l a s s e s a r e formations,

s a t u r a t e d formations and Schunck c l a s s e s . Genera l iza t ions of Hal l

subgroups a r e t h e dominating theme, and we begin wi th t h e theorems of Hal l

and Car t e r about t h e ex i s t ence and conjugacy of Hal l and Car t e r subgroups

( r e spec t ive ly ) i n f i n i t e so lvab le groups. i

A b a s i c no t ion i n r e c e n t work i s t h a t of covering subgroups of a

f i n i t e group, where t h e main r e s u l t i s t h a t if f i s a s a t u r a t e d formation

then every f i n i t e so lvab le group has f-covering subgroups and any two of

them a r e conjugate.

A more genera l no t ion than t h a t of covering subgroups i s t h a t of

p r o j e c t o r s ; however, i n t h e case of s a t u r a t e d formations t h e covering

subgroups and p r o j e c t o r s of any f i n i t e so lvab le group coinc ide and form

a s i n g l e conjugacy c l a s s . Moreover, t h e covering subgroups (p ro jec to r s )

f o r t h e formation of f i n i t e n i l p o t e n t groups a r e t h e C a r t e r subgroups.

In a d d i t i o n , pul l-backs exis . t f o r the f -pro jec . tors a s soc ia t ed with a

s a t u r a t e d formation f of f i n i t e so lvab le groups, and y i e l d s t h e

cons t ruc t ion of c e r t a i n formations.

A n a t u r a l ques t ion i s : which c l a s s e s F g i v e r i s e t o f - p r o j e c t o r s ?

The answer i s t h a t f o r Schunck c l a s s , which a r e more genera l than

s a t u r a t e d formations, every so lvab le group has &covering subgroups iff F

i s a Schunck c l a s s . In t h i s case F-covering subgroups a r e conjugate.

( i i i )

This i s now known t o be t r u e i f "F-covering subgroups" i s replaced by

''F-proj ectors", and we prove t h i s r e s u l t using p r o j e c t o r s .

In f a c t these theorems can be extended t o f i n i t e IT-solvable groups,

and i f F i s a IT-saturated formation (or a IT-Schunck c l a s s ) then every

f i n i t e IT-solvable group has f-covering subgroups (p ro jec to r s ) and any

two of them a r e conjugate.

The ex i s t ence of p r o j e c t o r s i n any f i n i t e group i s proved; however,

they may no t be conjugate and may no t coincide with t h e covering subgroups,

which might not a t a l l e x i s t , i n t h i s ca&. But i f F is a Schunck c l a s s

then F-projec tors and F-covering subgroups do coincide i n groups i n UF,

although t h e F-pro j e c t o r s need no t be con jugate .

Acknowledgement

I wish t o thank my s e n i o r supervisor Professor J . L . Berggren f o r

suggest ing the t o p i c and f o r h i s s i n c e r e and f r i e n d l y h e l p and guidance.

I would a l s o l i k e t o thank Professor N.R. R e i l l y and Professor C . Godsil

f o r t h e i r va luab le remarks and comments. I wish a l s o t o thank a l l

f a c u l t y and s t a f f members of the Mathematics and S t a t i s t i c s Department.

My thanks a l s o t o Cindy L i s t e r f o r typing t h i s manuscript .

NOTAT ION

P \ 9

< 1>

1

Aut (G)

A \ B

g i s an element of G .

elements of groups.

s e t s , groups.

c l a s s e s of groups.

H i s isomorphic with G .

H i s a subgroup of G .

H i s a proper subgroup of G .

H is a normal subgro y p o f G .

t h e order of t h e group G .

Index of t h e subgroup H i n t h e group G .

p d i v i d e s q.

t h e i d e n t i t y subgroup.

t h e i d e n t i t y element of a group.

t h e group of automorphisms of G .

= ( x : x E A and X B B).

= ( p : P i s a prime and p \ \ G I ) .

= g-lhg where g,h C G and G i s a group.

-1 -1 = x y xy.

= .

= ( g E G : hg = h f o r a l l h E G I - t h e c e n t r e of G .

= ( g E G : hg = h f o r a l l . h E H) - t h e c e n t r a l i z e r of

PI i n G .

= ( g E G : hg E H f o r a l l 'h E H I - t h e normalizer of

H i n G .

a group with every element having order a power of t h e

prime p.

a group with every element having order a power of p ,

where p € n, and n i s a f ixed s e t of primes.

a group with every element having order a power of p ,

where p is a prime not i n n , n is a f ixed s e t of

primes.

on (G) t h e l a r g e s t normal n-subgroup of G . H x K d i r e c t product of H and K .

Maximal subgroup : proper subgroup, n o 6 contained i n any g r e a t e r proper

subgroup.

Minimal normal subgroup of G : normal subgroup M # of G which does

no t conta in normal subgroups of G except and M.

( v i i )

TABLE OF CONTENTS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approval

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstrac t . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . Tab1 e of Contents

L i s t o f F i g u r e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 1.

Chapter 2 . P ro jec to r s and Formations . . . . . . . . . . . . . . . Chapter 3 . Pro jec to r s and S c h u n c k ~ a s s e s . . . . . . . . . . . . . Chapter 4. Covering Subgroups and Pro jec to r s i n F i n i t e

n-solvableGroups . . . . . . . . . . . . . . . . . . . Chapter 5. P ro jec to r s of F i n i t e Groups . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography

. ( i i )

( i i i )

(v)

. (v i ) ( v i i i )

( ix )

1

26

. 58

( v i i i )

LIST OF FIGURES

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F i g u r e 1 53

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 2 61

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F i g u r e 3 64

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F i g u r e 4 66

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F i g u r e 5 67 F i g u r e 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .