SUBGROUPS OF GL2(F`)

AARON LANDESMAN

1. INTRODUCTION

In this course, we will classify all subgroups of GL2(F`) for ` aprime. We start by recalling the definition of GL2(F`)

Definition 1.1. Let k be a field. The general linear group GL2(k) isthe set of matrices

{M ∈ Mat2×2(k) : M is invertible}More concretely, it is the set of matrices(

a bc d

)with a, b, c, d ∈ k such that det M := ad− bc 6= 0.

In this course, you will work out for yourself a classification of allsubgroups of GL2(F`). For this, we will introduce some importantsubgroups which come into play. We define the following subgroupsof matrices:

Definition 1.2. Any subgroup, which up to conjugation is of theform {(

a b0 c

): a, b, c ∈ F`, ac 6= 0

}is called a Borel subgroup.

Any subgroup, which up to conjugation is of the form{(a 00 c

): a, c ∈ F`, ac 6= 0

}is called a split Cartan subgroup.

Let ε ∈ F×` be a non-square (i.e., an element so that there is not ∈ F` with ε = t2). Then, any subgroup, which up to conjugation isof the form {(

a εbb a

): a, b ∈ F`, a2 − εb2 6= 0

}1

2 AARON LANDESMAN

is called a nonsplit Cartan subgroup.A subgroup is called Cartan if it is either a split Cartan or nonsplit

Cartan subgroup.

Example 1.3. As mentioned in Definition 1.2, the subgroups are onlydefined up to conjugacy. For example, consider a split Cartan sub-group C. By definition, there is some g ∈ GL2(F`) with C = gDg−1,for D the diagonal split Cartan

D :={(

a 00 c

): a, c ∈ F`, ac 6= 0

}.

In general, a the split Cartan C will take the form

C =

{(awx− cyz (a− c)wy(c− a)xz cwx− azy

): a, c ∈ F`, ac 6= 0

}for fixed x, y, z, w ∈ F` such that xw− zy 6= 0.

Exercise 1.4 (Optional). Verify that all split Cartans are indeed of thisform. 1

We are nearly ready to state the main classification theorem, butfirst we must introduce two groups related to GL2(F`).

Definition 1.5. Define special linear group to be the matrix groupSL2(k) ⊂ GL2(k) to be

{M ∈ GL2(k) : det M = 1} .

Define the subgroup of scalar matrices

Z :={(

a 00 a

): a 6= 0

}.

Finally, define the projective linear group to be the quotient group

PGL2(k) := GL2(k)/Z.

Definition 1.6. Recall that for G a group, a maximal subgroup H ⊂G is a subgroup H 6= G so that for any subgroup K with H ⊂ K ⊂ G,either H = K or G = K.

The main theorem we will prove is the following:

Theorem 1.7. Suppose G ⊂ GL2(F`) is a maximal subgroup. Then,either

(1) G contains SL2(F`)(2) G is a Borel subgroup(3) G is the normalizer of a Cartan subgroup(4) the image of G in PGL2(F`) is isomorphic to A4, S4, or A5.

SUBGROUPS OF GL2(F`) 3

The final statement of the theorem we prove will be slightly morerefined than this, see Theorem 7.2, and will give a concrete descrip-tion of what all possible subgroups look like.

4 AARON LANDESMAN

2. SUBGROUPS OF GL2(F2)

To start, we tackle by hand the case that ` = 2.

Exercise 2.1. Show that # GL2(F2) = 6 by explicitly writing out thesix elements in GL2(F2).

Exercise 2.2. Show GL2(F2) ' S3. 2

Exercise 2.3. Conclude from the previous problem that the all sub-groups of GL2(F2) other than id and GL2(F2) are either isomorphicto Z/3Z or Z/2Z.

For the remainder of the course, we shall assume ` > 2 withoutfurther comment.

SUBGROUPS OF GL2(F`) 5

3. SUBGROUPS OF ORDER DIVISIBLE BY `

To start, we will classify all subgroups of order divisible by `,showing they are either contained in a Borel subgroup or else containSL2(F`). This is carried out in Exercise 3.13.

3.1. Counting the number of 1-dimensional subspaces.

Exercise 3.1. Show that if V, W ⊂ F2` are two distinct one dimen-

sional subspaces of F2` , then V ∩W = {0}.

Exercise 3.2. Using Exercise 3.1, show that every nonzero elementof the 2-dimensional vector space F2

` lies in a unique 1-dimensionalsubspace of F2

` .

Exercise 3.3. Show there are precisely `+ 1 1-dimensional subspacesof F2

` .3

3.2. Normal form for order ` elements. .

Exercise 3.4. Suppose M ∈ GL2(F`) has order `. Show that M hasan eigenvector. 4

Exercise 3.5. Show that any matrix M of order ` does not have twoindependent eigenvectors. 5

Exercise 3.6. Suppose that M ∈ GL2(F`) is a matrix with one butnot two distinct eigenvalues. Show that x2 − tr(M)x + det(M) =

(x− λ)2 , for λ the unique eigenvalue of M. 6

Definition 3.7. For k a field, we say a matrix M ∈ GL2(k) is a scalarif it is of the form (

a 00 a

)for a ∈ k, a 6= 0. We say a matrix is a non-scalar matrix if it is not ofthe above form.

Exercise 3.8. Suppose that M ∈ GL2(F`) is a non-scalar matrix withone but not two eigenvalues. Let v be an eigenvector with eigenvalueλ. Show that M−λ · id has a nontrivial kernel but is not the 0 matrix.Conclude dim im(M− λ id) = 1 and in fact im (M− λ id) = Span v.

Exercise 3.9. Let M ∈ GL2(F`) be a non-scalar matrix with one butnot two eigenvalues. Suppose v is an eigenvector of M with eigen-value λ.

(1) Using that im(M− λ · id) = Span v, show there is some vec-tor w with (M− λ · id)w = v.

6 AARON LANDESMAN

(2) Use the preceding part to show that, up to conjugation, wemay write any non-scalar matrix M with a unique eigenvaluein the form (

a 10 a

)for a ∈ F`. 7

Exercise 3.10. Using Exercise 3.9 and show that if M has order `,then, up to conjugation, M is of the form(

1 10 1

).

Conclude that every such M is contained in a Borel subgroup.

3.3. Completing the classification of subgroups containing an ele-ment of order `.

Exercise 3.11. Show that the two elements

S :=(

1 10 1

)and U :=

(1 01 1

)generate SL2(F`) in the following steps.

(1) Recall SL2(Z) is the set of 2× 2 matrices with integer entries,whose determinant is 1. Show that it suffices to show S andU, when considered as matrices over the integers (and notwith entries in F`), generate SL2(Z). 8

Remark 3.12. For the remainder of the problem, we considerS and U as matrices with integer entries.

(2) Show that the element

T :=(

0 −11 0

)∈ SL2(Z)

can be expressed as

T = S−1US−1.

Argue it suffices to show S and T generate SL2(Z).(3) Show that

T(

a bc d

)=

(−c −da b

)Sn

(a bc d

)=

(a + nc b + nd

c d

)

SUBGROUPS OF GL2(F`) 7

(4) Show via multiplying by suitable powers of S and T that if Sand T generate matrices with lower left hand corner 0, thenthey generate all matrices. 9

(5) Show that all matrices M ∈ SL2(Z) with lower left hand cor-ner equal to 0 indeed lie in the subgroup generated by S andT. 10

Exercise 3.13. Suppose that G ⊂ GL2(F`) contains two order ` ele-ments, neither of which is a power of the other. Argue using Exer-cise 3.10 that, up to conjugating G by some element of GL2(F`), onecan assume that G contains the two elements(

1 10 1

)and

(1 01 1

).

Conclude from Exercise 3.11 that G contains SL2(F`).

8 AARON LANDESMAN

4. JORDAN NORMAL FORM

Next, we work out a description of matrices in GL2(F`) known ina broader context as “Jordan normal form.”

Definition 4.1. Let M be an n× n matrix over a field L. Let v1, . . . , vnbe a maximal collection of linearly independent eigenvectors over K.We say that a matrix M has all eigenvectors defined over K if for anyfinite extension L/K, {v1, . . . , vn}, considered as eigenvectors over Lremains a maximal collection of linearly independent eigenvectors.

Exercise 4.2. Show that any matrix M ∈ GL2(F`), when consideredas a matrix over F`2 has all eigenvectors defined over F`2 . 11

Exercise 4.3. By the previous exercise, for M ∈ GL2(F`), M has aneigenvector v ∈ F2

`2 with eigenvalue λ over F`2 .(1) Suppose that M has a unique eigenvalue when considered as

an element of GL2(F`2). Show that either M is a scalar matrixor else has order divisible by `. 12

(2) Suppose that M has two distinct eigenvalues. Suppose thatthere are two independent vectors v and w ∈ F2

`2 which areboth eigenvectors for M. Show that M, when considered as amatrix over F`2 can be written in the form(

a 00 b

)for a, b ∈ F`2 . (This part does not use anything specific to F`2 ,and holds equally well over any field.)

SUBGROUPS OF GL2(F`) 9

5. ELEMENTS OF ORDER PRIME TO ` LIE IN A CARTAN

Our next main goal is to show that every element of order primeto ` lies in a Cartan subgroup. For this, we will start with a geometriccharacterization of split Cartan and nonsplit Cartan subgroups.

5.1. Geometric characterization of split Cartans. To start, we iden-tify split Cartans as the stabilizers of two lines in F2

` .

Exercise 5.1. Suppose that G ⊂ GL2(F`) is a split Cartan subgroup.Show that there are two lines L1, L2 ⊂ F2

` (i.e., Li are one dimensionalsubspaces of F2

`) with G · L1 = L1 and G · L2 = L2. Conversely, giventwo lines, L1 and L2, show that there is a unique Cartan subgroup Gwith G · L1 = L1 and G · L2 = L2.

Exercise 5.2. Show that as a group, any split Cartan is isomorphic toF×` ×F×` .

5.2. Geometric characterization of nonsplit Cartans. We next iden-tify nonsplit Cartans as the stabilizers of two lines, only defined overF`2 .

Exercise 5.3. Fix some σ ∈ F`2 with σ2 = ε, for ε ∈ F×` a non-square in F×` , as in the definition of nonsplit Cartan subgroups, Def-inition 1.2.

Show that there are two lines defined over F`2 but not over F`

which are fixed by any nonsplit Cartan subgroup. 13

Exercise 5.4. In this exercise, we show that any nonsplit Cartan isprecisely the elements of GL2(F`) which stabilizes each of two linesdefined over F`2 but not over F`.

(1) Choose an element σ ∈ F`2 with σ2 = ε, for ε a non-square asin Definition 1.2. Show that (1, σ) form a basis of F`2 over F`,which yields an isomorphism F`2 ' F2

` .(2) Show that under the identification above, multiplication by

an element of the form x · 1 + y · σ ∈ (F2`)× acts on F2

` withbasis (1, σ) by the matrix(

x εyy x

)14

(3) Using the above, show that a nonsplit Cartan can be identi-fied with F×

`2 via the action of F×`2 on F2

` .

10 AARON LANDESMAN

(4) Given an element M ∈ GL2(F`), suppose that M has twoeigenvalues defined over F`2 but none over F`, and that thesetwo eigenvalues correspond to eigenvectors fixing two linesL1 and L2 defined over F`2 . Show that the two eigenvalues areconjugate under the nontrivial automorphism (i.e., frobenius)of F`2 over F`, and therefore that the action on one of the linesdetermines the action on the other.

(5) Conclude that there are precisely #F×`2 elements of GL2(F`)

which fix the subspaces L1 and L2 (realized as stabilizers ofsome M as in the previous part) defined over F`2 .

(6) Conclude that any nonsplit Cartan is precisely the stabilizerof the two lines L1 and L2 defined over F`2 but not over F`.

5.3. Intersections of Cartans. We next show that any two Cartansubgroups intersect precisely in the scalar matrices.

Exercise 5.5. Show that any non-scalar element M ∈ GL2(F`) has atmost two lines L with ML ⊂ L.

Exercise 5.6. Conclude that the intersection of any two split Cartansubgroups is precisely the scalar matrices. 15

Exercise 5.7. Conclude that the intersection of any split Cartan sub-group with any nonsplit Cartan subgroup is precisely the scalar ma-trices. 16

Exercise 5.8. Show that the intersection of any two nonsplit Cartansubgroups is precisely the scalar matrices.

5.4. Showing every non-scalar lies in a unique Cartan. To concludethis section, we show any non-scalar element of order prime to ` liesin a unique Cartan subgroup.

Exercise 5.9. Suppose M ∈ GL2(F`) has order prime to ` and is not ascalar. Show that M has two distinct eigenvalues using Exercise 3.9.

Exercise 5.10. Show that any non-scalar element M with order primeto ` lies in a unique Cartan subgroup. 17

SUBGROUPS OF GL2(F`) 11

6. NORMALIZERS OF CARTAN SUBGROUPS

We now compute the normalizer of a Cartan subgroup. We beginby recalling the definition of normalizer.

Definition 6.1. Let H ⊂ G be a subgroup. The the normalizer of Hin G is the set of elements g ∈ G so that gHg−1 = H.

Exercise 6.2. For H ⊂ G a subgroup, show that the normalizer of Hin G is a subgroup of G.

Exercise 6.3. Let G ⊂ GL2(F`) be a Cartan subgroup (either split ornonsplit). Let L1 and L2 be the two lines (possibly only defined overF`2 in the nonsplit case) associated to G via Exercise 5.1 for the splitcase and Exercise 5.4 for the nonsplit case.

(1) Show that any element of the normalizer of G in GL2(F`)must send L1 ∪ L2 to L1 ∪ L2.

(2) Show that any element of the normalizer of G in GL2(F`)which does not lie in G must swap L1 and L2.

(3) Show that any Cartan subgroup has index two in its normal-izer.

The following exercise provides a concrete description of the nor-malizers, though technically speaking it is not needed for the classi-fication. Feel free to skip it if you would like.

Exercise 6.4 (Bonus, not needed for classification). Show that thenormalizer of a split Cartan can be explicitly realized as those ele-ments of the form (

a 00 b

)or

(0 ab 0

)for a, b 6= 0, a, b ∈ F2

` . Similarly, show that the normalizer of a non-split Cartan can explicitly be realized as those elements of the form(

x εyy x

)or

(x εy−y −x

)for x, y 6= 0, x, y ∈ F2

` .

Let φ : GL2(F`) → PGL2(F`) denote the quotient map, with PGLdefined in Definition 1.5.

Exercise 6.5. Suppose G ⊂ GL2(F`) is a subgroup with φ(G) ⊂PGL2(F`) cyclic and #G is prime to `. Show that G is contained in aCartan subgroup. 18

12 AARON LANDESMAN

Exercise 6.6. Suppose G ⊂ GL2(F`) is a subgroup so that φ(G) ⊂PGL2(F`) is a dihedral group with #G prime to `. Show that G iscontained in the normalizer of a Cartan subgroup via the followingsteps:

(1) Reduce to showing φ−1(φ(G)) is contained in the normalizerof a Cartan subgroup.

(2) Show there exists H ⊂ φ(G) a cyclic subgroup of φ(G) oforder #φ(G)/2 with φ(G) the normalizer of H in φ(G).

(3) Show that φ−1(φ(G)) is the normalizer of φ−1(H) in φ−1(φ(G)).(4) Use Exercise 6.5 to show φ−1(H) is contained in a Cartan.(5) Conclude φ−1(φ(G)) is contained in the normalizer of a Car-

tan.

SUBGROUPS OF GL2(F`) 13

7. FINISHING THE CLASSIFICATION, MODULO EXCEPTIONALSUBGROUPS

We are now ready to complete the classification, modulo a techni-cal computation on subgroups of PGL2(F`), which you can return toprove later if you have time.

We first state this auxiliary result, and then use it to deduce themain classification.

Theorem 7.1. Let H ⊂ PGL2(F`) be a subgroup of order prime to `. If His not cyclic or dihedral, then H is either isomorphic to A4, S4 or A5 (withSn the symmetric group on n letters and An ⊂ Sn the alternating group onn letters).

Using this, we now prove our main result, which is a slightly up-graded form of Theorem 1.7.

Theorem 7.2. Suppose G is a subgroup of GL2(F`) and let H denote theimage of G in PGL2(F`). If G has order divisible by ` then either

(1) G is contained in a Borel subgroup(2) G contains SL2(F`).

If G has order prime to ` then either(i) H is cyclic and G is contained in a Cartan subgroup

(ii) H is dihedral and G is contained in the normalizer of a Cartan sub-group, but not in a Cartan subgroup.

(iii) H is isomorphic to A4, S4, or A5.

Exercise 7.3. Deduce Theorem 1.7 from Theorem 7.2.

Exercise 7.4. Prove Theorem 7.2 in two steps (assuming Theorem 7.1):(1) Show that any subgroup of G with order divisible by ` is

either contained in a Borel or contains SL2(F`) using Exer-cise 3.13. 19

(2) Show that if G has order prime to `, it satisfies one of cases(i), (ii), or (iii) using Theorem 7.1 together with Exercise 6.5and Exercise 6.6.

Exercise 7.5. Give a finer description of (2) of Theorem 7.2 by show-ing that subgroups G ⊂ GL2(F`) containing SL2(F`) are in bijectionwith subgroups of F×` by sending

G 7→{

a ∈ F×` : there exists g ∈ G, det g = a}

.20

14 AARON LANDESMAN

8. CLASSIFICATION OF SUBGROUPS OF PGL2(F`)

To complete the proof of our classification of subgroups of GL2(F`),it suffices to prove Theorem 7.1, classifying subgroups of PGL2(F`).

8.1. Restricting the possibilities. We now want to find which sub-groups of PGL2(F`) can arise that are neither cyclic nor dihedral.

Definition 8.1. For G a finite group, a maximal cyclic subgroup isa subgroup H ⊂ G (possibly equal to G) which is cyclic and is notcontained in any strictly larger cyclic subgroup.

Exercise 8.2. Let G ⊂ PGL2(F`) be a subgroup. Show that any twodistinct maximal cyclic subgroups of G intersect only in the identity.21

Exercise 8.3. Let G ⊂ PGL2(F`) be a subgroup. Let H ⊂ G be amaximal cyclic subgroup.

(1) Show that the normalizer of H in G is either equal to H or elsecontains H as a subgroup of index 2. 22

(2) In the first case that H is its own normalizer, show that thereare #G/#H maximal cyclic subgroups of G conjugate to H. 23

(3) In the second case that H has index 2 in its normalizer, showthat there are (#G/2)

#H maximal cyclic subgroups of G conjugateto H.

We now introduce some notation which will be used heavily through-out the following exercises:

Definition 8.4. Let G ⊂ PGL2(F`) be a subgroup. Let H1, . . . , Hr ⊂G denote maximal cyclic subgroups of G so that every element of Gis conjugate to an element in some Hr, and no two Hr’s are conjugate.Let di := #Hi. Let fi be 1 if the normalizer of Hi in G is Hi and letfi be 2 if Hi has index 2 in its normalizer in G. (These are the onlypossibilities using Exercise 8.3).

The following exercise is the key to completing the classification.

Exercise 8.5 (Key exercise). Using the notation of Definition 8.4, showthat

(8.1) #G = 1 +r

∑i=1

(di − 1)#Gfidi

SUBGROUPS OF GL2(F`) 15

with fidi ≤ #G for all i (recall r is the number of subgroups H1, . . . , Hrdefined in Definition 8.4). Rewrite the above equation in the form

(8.2) 1−r

∑i=1

di − 1fidi

=1

#G.

24

Exercise 8.6. Show that r ≤ 3 above. 25

So, it now remains to deal with the cases r = 1, 2, and 3 separately.

Exercise 8.7. Rule out the case r = 1 as follows:(1) Show that if r = 1 and f1 = 2 then #G in Equation 8.2 cannot

be an integer.(2) Show that if r = 1 and f2 = 1 then G is a cyclic group.

Exercise 8.8. Show that if r = 2 then if G ⊂ PGL2(F`) is a subgroupwhich is neither cyclic nor dihedral, then, up to permutation of thedi and fi, we must have d1 = 3, d2 = 2, f1 = 1, f2 = 2 as follows:

(1) Rule out the case f1 = f2 = 1 by showing that

1#G

> 1− d1 − 1d1

− d2 − 1d2

.

(2) Rule out the case f1 = f2 = 2 by showing that in this case onecan rearrange Equation 8.2 to conclude

2#G

=1d1

+1d2

.

Use the condition of Exercise 8.5 that fidi ≤ #G to show 2#G ≤

1d1

to obtain a contradiction.(3) Finally, deal with the case f1 = 1, f2 = 2, in the following

steps:(a) Show in this case that

1#G

=1d1

+1

2d2− 1

2.

(b) Use that d2 ≥ 2 to conclude that d1 < 4. So, either d1 = 2or d1 = 3.

(c) If d1 = 2, show that #G = 2d2 and conclude that G isdihedral.

(d) If d1 = 3, show that we must have d2 = 2 and #G = 12.

16 AARON LANDESMAN

Exercise 8.9. Show that if r = 3 then if G ⊂ PGL2(F`) is a subgroupwhich is neither cyclic nor dihedral, then, up to permutation of thedi, f1 = f2 = f3 = 2, d1 = 2, d2 = 3, and d3 ∈ {3, 4, 5} as follows:

(1) Show that f1 = f2 = f3 = 2 by ruling out the possibility thatf1 = 1.

(2) Show that Equation 8.2 can be rewritten in the case f1 = f2 =f3 = 2 as

1 +2

#G=

1d1

+1d2

+1d3

.

(3) Show that we cannot have 3 ≤ d1 ≤ d2 ≤ d3, so we mayassume d1 = 2.

(4) Show that if d1 = 2, then it is not possible to have 4 ≤ d2 ≤ d3.Therefore, we may assume that d2 ∈ {2, 3}.

(5) Show that if d1 = d2 = 2 then #G = 2d3 and G is dihedral.(6) Show that if d1 = 2 and d2 = 3, then d3 ∈ {3, 4, 5}, and

in these three cases we have #G = 12, #G = 24, #G = 60respectively.

Exercise 8.10. Conclude that any subgroup of PGL2(F`) which is notcyclic or dihedral must have one of the following sets of associatedinvariants:

(1) r = 2, d1 = 3, d2 = 2, f1 = 1, f2 = 2 and #G = 12(2) r = 3, d1 = 2, d2 = 3, d3 = 3, f1 = f2 = f3 = 2 and #G = 12.(3) r = 3, d1 = 2, d2 = 3, d3 = 4, f1 = f2 = f3 = 2 and #G = 24.(4) r = 3, d1 = 2, d2 = 3, d3 = 5, f1 = f2 = f3 = 2 and #G = 60.

8.2. Determining the exceptional groups.

Exercise 8.11. Show that in the case r = 2 and d1 = 3, d2 = 2, f1 =1, f2 = 2 that the subgroup G ' A4. You may use the fact that A4is the only non-commutative group of order 12 with all elements oforder less than 4.

Exercise 8.12. Show that the case r = 3, d1 = 2, d2 = 3, d3 = 3 andf1 = f2 = f3 = 2 cannot occur. 26

Exercise 8.13. Show that in the case r = 3, d1 = 2, d2 = 3, d3 = 4and f1 = f2 = f3 = 2, we have G ' S4. You may use the factthat S4 is the only group of order 24 with four 3-Sylow subgroupsand no element of order 6 (see https://math.stackexchange.com/

q/534718 for a proof).

Exercise 8.14. Show that in the case r = 3, d1 = 2, d2 = 3, d3 = 5and f1 = f2 = f3 = 2, we have G ' A5. You may use that A5 is

SUBGROUPS OF GL2(F`) 17

the only group of order 60 with six 5-Sylow subgroups. See http:

//www.math.toronto.edu/alfonso/347/Groups60.pdf.

8.3. Completing the proof.

Exercise 8.15. Complete the proof of Theorem 7.2 using the exercisesin subsection 8.2 to deal with the four cases of Exercise 8.10.

18 AARON LANDESMAN

9. BONUS MATERIALS

We have completed the proof of our main theorem Theorem 1.7,but there are still some remaining related exercises.

9.1. Discriminants, determinants, and traces.

Definition 9.1. Given a matrix M ∈ GL2(F`), define the discrimi-nant to be

∆(M) := (tr(M))2 − 4 det(M).

Concretely, if

M =

(a bc d

)then

∆(M) = (a + d)2 − 4 (ad− bc) .

Exercise 9.2. Show that an element M ∈ GL2(F`) with ∆(M) 6= 0 iscontained in a unique Cartan subgroup. 27

Exercise 9.3. Suppose that ∆(M) 6= 0. Show that M is containedin a split Cartan subgroup if and only if ∆ is a square mod` and Mis contained in a nonsplit Cartan subgroup if and only if ∆ is not asquare mod`.

Exercise 9.4. Let G ⊂ GL2(F`) be a Cartan subgroup and let N de-note its normalizer. Show in two ways that every element of N \ Ghas trace 0.

(1) As a concrete approach use Exercise 6.4.(2) More abstractly, give a proof using the geometric description

of element of N \G as permuting the two lines which G fixes.

9.2. A criterion for containing all of GL2(F`). We now prove thefollowing criterion for a subgroup of GL2(F`) to be equal to all ofGL2(F`).

Theorem 9.5. Suppose ` ≥ 5 and G ⊂ GL2 (F`) is a subgroup so that(i) G has an element s with ∆(s) a nonzero square mod` and tr(s) 6=

0.(ii) G has an element s′ with ∆(s′) a non-square mod ` and tr(s′) 6= 0.

(iii) G has an element s′′ with u := tr(s′′)2

det(s′′) so that u /∈ {0, 1, 2, 4} andfurther that u2 − 3u + 1 6= 0.

Then, G contains SL2(F`).

SUBGROUPS OF GL2(F`) 19

Before proving the theorem, we deduce some corollaries:

Exercise 9.6. Show that if G ⊂ GL2(F`) satisfies the conditions ofTheorem 9.5 and the determinant map G → F×` is surjective, thenG = GL2(F`).

Exercise 9.7. Suppose that G ⊂ GL2(F`) is a subgroup with theproperty that for every t ∈ F`, d ∈ F×` there is some s ∈ G withtr s = t and det s = d. If ` ≥ 5, show that G = GL2(F`).

We now proceed to outline a proof of Theorem 9.5 in a series ofexercises.

Proof of Theorem 9.5. Suppose SL2(F`) is not contained in G. Then,by Theorem 1.7, either G is contained in a Borel, the normalizer ofa Cartan, or its image in PGL2(F`) contains one of the exceptionalgroups A4, S4, or A5.

Exercise 9.8. Show that G is not contained in the normalizer of aCartan using Exercise 9.4, Exercise 9.3 together with conditions (i)and (ii) of the theorem.

Exercise 9.9. Rule out the possibility that G is contained in a Borelsubgroup, using condition (ii).

By the above, it suffices to show that the image in PGL2(F`) of Gis not one of A4, S4 or A5.

Exercise 9.10. Rule out that this last possibility via the followingsteps.

(1) Let s ∈ GL2(F`) have eigenvalues λ, τ over GL2(F`2). If theorder of φ(s) ∈ PGL2(F`) is r, show that λ

τ is a primitive rthroot of unity. In particular,(

λ

τ

)r= 1.

(2) In the notation above, show that

tr(s)2

det(s)=

λ

τ+ 2 +

τ

λ= ζr + 2 + ζ−1

r ,

for ζr a primitive rth root of unity.

(3) Show that if r ∈ {1, 2, 3, 4} then u := tr(s)2

det(s) ∈ {4, 0, 1, 2} and if

r = 5 then u2 − 3u + 1 = 0.(4) Show that every element of A4, S4, and A5 has order at most

5.

20 AARON LANDESMAN

(5) Conclude that φ(s′′) is not contained in a subgroup of PGL2(F`)isomorphic to A4, S4, or A5.

�

9.3. Finite subgroups of PGL2(k), prime to the characteristic. Fi-nally, generalizing the above, we can use the above techniques toclassify precisely which subgroups appear as finite subgroups of PGL2(k)for any field k. Furthermore, we can realize all the subgroups as ap-pearing in the case k = C.

Exercise 9.11 (Involved Exercise). Let k be any field. Show that anyfinite subgroups of PGL2(k) of order prime to char k are either iso-morphic to A4, S4, A5, cyclic, or dihedral. 28

Exercise 9.12. Show that A4, S4, A5, as well as all dihedral groupsand cyclic groups appear as finite subgroups of PGL2(C). Deduce acomplete list of isomorphism classes of finite subgroups of PGL2(C).29

NOTES 21

NOTES

1Hint: Conjugate by g with

g−1 =

(x yz w

).

2Hint: One can do this mundanely, by writing out all six elements but for a moreaesthetically pleasing solution, consider GL2(F2) as acting on the 2-dimensionalvector space V over F2 with nonzero elements e1, e2, e1 + e2. Identify the action ofGL2(F2) on e1, e2, e1 + e2 with the permutation action of S3 on the three elementset {e1, e2, e1 + e2}.

3Hint: Relate the number of 1-dimensional subspaces to the number of nonzeroelements.

4Hint: To show M has an eigenvector, show M fixes a line. Use that there are`+ 1 lines by Exercise 3.3 and that M has order `.

5Hint: If M has two independent eigenvectors, show that up to conjugation itcan be written as a diagonal matrix.

6Hint: Use that the roots of the characteristic polynomial are precisely the eigen-values.

7Hint: Use that if a matrix has an eigenvector with eigenvalue a, one can changebasis so that it is upper triangular with lower right hand corner entry equal to a.

8Hint: Show SL2(Z)→ SL2(F`) is surjective by taking some M ∈ SL2(F`) andchoosing some lift (

a bc d

)of M as a 2× 2 matrix over Z. Show one can assume a and b are relatively primeand then add appropriate multiples of p to c and d.

9Hint: Start with some matrix

M =

(a bc d

)with c 6= 0. Use the two operations from the preceding part to apply the Euclideandivision algorithm over Z to a and c to arrange that c = 0.

10Hint: Show that any matrix with lower left hand corner 0 is either of the form(1 a0 1

)or

(−1 a0 −1

).

Use that S2 = − id to assume it is of the first form.11Hint: Show that a 2 × 2 matrix M has all eigenvectors defined over a field

if and only if the polynomial λ2 − tr(M)λ + det(M) factors as a product of twolinear factors. Show that if this polynomial does not factor over F` then F`2 'F`[x]/(x2 − tr(M)x + det(M)), in which case it factors over F`2 .

12Hint: Use Exercise 3.9.13Hint: Choosing a nonsplit Cartan as in Definition 1.2, identify the lines with

the eigenvectors of any non-scalar element of the subgroup, and show that allelements in the nonsplit Cartan have these same eigenvectors.

14Hint: What is (x · 1 + y · σ) · 1 and (x · 1 + y · σ) · σ?15Hint: Use Exercise 5.1 and Exercise 5.5.

22 NOTES

16Hint: Use Exercise 5.5 together with both Exercise 5.5 and Exercise 5.1 to showany such element would stabilize at least four distinct lines.

17Hint: By the previous exercise M has two eigenvalues defined over F`2 . Ifthese eigenvalues are defined over F`, show that M stabilizes two lines L1 and L2defined over F` and conclude that it lies in the unique split Cartan via Exercise 5.1.If M has two eigenvalues defined over F`2 but not over F`, show M lies in a uniquenonsplit Cartan using Exercise 5.4.

18Hint: Choose a generator g ∈ φ(G), and choose some g ∈ G mapping to g.Show that G is contained in the group generated by g together with the scalarmatrices. Show that g lies in a Cartan subgroup, using Exercise 5.10.

19Hint: If G has some element M of order `, use Exercise 3.10 to show ` lies ina Borel subgroup B. Show that a conjugate of M by an element not in B yieldsanother element of order ` which is not a power of M.

20Hint: The inverse sends a subgroup H ⊂ F×` to those elements of GL2(F`)with determinant lying in H.

21Hint: Use that the preimages of these subgroups in GL2(F`) are contained indistinct Cartans, and hence intersect only in scalar matrices by Exercise 5.10.

22Hint: Use the analogous result on GL2(F`) applied to the preimage of H inGL2(F`).

23Hint: Consider the conjugation action of G on the quotient G/H, and use theorbit stabilizer lemma to determine the number of conjugates of H.

24Hint: Write G as a union of the conjugates of the subgroups H1, . . . , Hr, anduse inclusion-exclusion via Exercise 8.2.

25Hint: Show di−1fidi≥ 1/4.

26Hint: Show that for i, j ∈ {1, 2, 3},#G ≥ di(dj − 1) + di(dj − 1) + 1.

Use this to show such a group cannot occur.27Hint: Show that ∆(M) 6= 0 if and only if M has no repeated eigenvalues over

F`2 using the characteristic polynomial of M and the quadratic formula.28Hint: The entire proof given in section 8 goes through in nearly the same way

for an arbitrary field k, using the key exercise Exercise 8.5. The main obstacleis to show that Exercise 8.2 and Exercise 8.3 generalize appropriately. These canbe proven similarly to they way they were shown over F`, by considering theeigenvalues of any given matrix.

29Hint: For the exceptional cases, use the stereographic projection to identifyP1

C := C2/C× with S2 ⊂ R3 and consider the automorphism groups of the pla-tonic solids.