On Certain Subgroups of E8(2) and Their Brauer Character Tables

ON CERTAIN SUBGROUPS OF E8(2) AND THEIR BRAUER CHARACTER TABLES

A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science and Engineering

2018

Peter J Neuhaus School of Mathematics Contents

Abstract 5

Declaration 6

Acknowledgements 8

1 Introduction 9

2 Background Material 16 2.1 MaximalSubgroupsofAlgebraicGroups ...... 19

3 Preliminary Results 23

3.1 E8(2)...... 23 3.1.1 Maximal Subgroups ...... 23

3.1.2 Elements of E8(2) ...... 26 3.2 Brauer Characters ...... 32 3.2.1 Conway Polynomials ...... 33 3.2.2 Brauer Character of L(G)...... 34 3.2.3 Feasible Decompositions ...... 35 3.3 Irreducible Modules of Groups of Lie Type ...... 36 3.3.1 DimensionsofHighestWeightModules ...... 38 3.4 Determining Maximality ...... 39

4 Proof of Theorems 1.2 and 1.3 45

4.1 L2(7)...... 45

2 4.2 L2(11) ...... 46

4.3 L2(13) ...... 46

4.4 L2(17) ...... 46

4.5 L2(25) ...... 47

4.6 L2(27) ...... 48

4.7 L4(3)...... 49

4.8 U3(3)...... 49

4.9 G2(3)...... 49

4.10 M11 ...... 49

4.11 M12 ...... 50

4.12 L3(8)...... 50

4.13 F4(2)...... 50 2 4.14 F4(2)0 ...... 50 3 4.15 D4(2)...... 51

4.16 L3(3)...... 51 4.16.1 Case (ii) ...... 51 4.16.2 Case (i) ...... 52

4.17 L3(4)...... 56

4.18 U3(8)...... 57 4.18.1 Case (ii) ...... 57 4.18.2 Case (i) ...... 58

4.19 U3(16) ...... 61

5 Proof of Theorem 1.4 62 5.1 Methodology ...... 62

5.2 L2(128) ...... 66

5.3 L2(32) ...... 67 5.3.1 P(1234) ...... 67 5.3.2 P(1345) ...... 68 5.3.3 P(2456) ...... 68 5.3.4 P(3456) ...... 69 5.3.5 P(4567) ...... 69

3 5.3.6 P(5678) ...... 70

6 Brauer Character Tables 71

7 Further Work 123

Bibliography 124

APrograms 130 A.1 FeasibleCharacterCode ...... 130 A.2 IndecomposableModuleCode ...... 135

N A.3 L2(2 )code...... 138

B Supplementary Code 149

B.1 E8(2) setup ...... 149

4 The University of Manchester

Peter J Neuhaus Doctor of Philosophy On Certain Subgroups of E8(2) and their Brauer Character Tables December 4, 2018

For the exceptional group of Lie type E8(2) a maximal subgroup is either one of a known set or it is almost simple. In this thesis we compile a complete list of almost simple groups that may have a maximal embedding in E8(2) and in many cases it is proved that such an embedding does not exist. For the groups L2(32) and L2(128) we go further and ﬁnd all conjugacy classes of their embeddings in E8(2). Extensive use is made of the theory of Brauer characters and modular representation theory, and as such include Brauer character tables in characteristic 2 for many small rank simple groups. The work in this thesis relies heavily on the computer package Magma and includes a collection of useful procedures for computational group theory. The results presented are the author’s contribution to the ongoing attempt to classify the maximal subgroups of E8(2).

5 Declaration

No portion of the work referred to in the thesis has been submitted in support of an application for another degree or qualiﬁcation of this or any other university or other institute of learning.

6 Copyright Statement

i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the “Copyright”) and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes.

ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trade marks and other intellectual property (the “Intellectual Property”) and any reproductions of copyright works in the thesis, for example graphs and tables (“Reproductions”), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. iv. Further information on the conditions under which disclosure, publication and com- mercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see http://documents.manchester.ac.uk/DocuInfo.aspx?DocID=487), in any relevant Thesis restriction declarations deposited in the University Library, The Univer- sity Library’s regulations (see http://www.manchester.ac.uk/library/aboutus/regulations) and in The University’s Policy on Presentation of Theses.

7 Acknowledgements

First and foremost, I would like to thank Peter Rowley, for his guidance throughout my time at Manchester, his invaluable expertise and mathematical advice, as well as his patience and understanding. Without him this thesis would certainly not exist.

I also wish to thank the other members of my research group; John Ballantyne for sharing his expertise in Magma so clearly, Jamie Phillips and David Ward for showing me the ropes and making me feel so welcome when I was just starting, and Alexander McGaw, for being a constant friend throughout, willing to be a sounding board for new ideas and frustrations alike.

IwouldliketothankMatthewGwynneforhisfriendshipandhelpinnavigating the stresses that a PhD entails.

Finally, I would like to thank my parents, without whose unwavering support and belief I could not have achieved any of what I have done.

8 Chapter 1

Introduction

When studying finite groups and their properties there is a natural desire to obtain classifications of the objects in question, and in the case of simplicity this has been achieved by the classification of finite simple groups. This remarkable result was the work of countless mathematicians over more than a century, starting with Galois in 1830 and generally being considered to be finally complete with two volumes by As- chbacher and Smith in 2004 [5]. Whilst this classification provides constructions for all the groups, for example see Wilson [58], there is still much that is not known about their properties.

For almost as long as they have been studied, the structure of the maximal subgroups of these ﬁnite simple groups has been of great interest, and whilst a complete classiﬁcation has not been achieved much progress has been made. In 1901 the max-

a imal subgroups of L2(q), for q = p a power of a prime, were classiﬁed by Dickson

[20], and this was followed up by the classification of maximal subgroups for L3(q) by Mitchell [53], for q odd, and Hartley [29], for q even. Since then, after extensive study, the maximal subgroups of the finite simple groups have either been classified or categorised with the exception of the Monster group.

The categorising of maximal subgroups was ﬁrst achieved by the O’Nan-Scott theorem, initially proved independently by O’Nan and Scott; however, due to the importance of the result, there are now many proofs, for example by Aschbacher [3]. In this theorem it is proved that any maximal subgroup of a symmetric group is one

9 CHAPTER 1. INTRODUCTION 10 of five well understood types, or is almost simple. This is a common theme in the classification of maximal subgroups and is mirrored by Liebeck, Praeger and Saxl in their classification of the maximal subgroups of the finite (simple) alternating groups [40]. In 1984 Aschbacher [2] proved a similar theorem for the finite simple groups of classical Lie type.

The sporadic simple groups can have full classifications as they are finite in number. They have been tackled in a more piecemeal fashion across many papers by several authors, although the majority can be attributed to Wilson. For the classifications of maximal subgroups of sporadic simple groups one can read the following papers; M24

[15] [19], J1 [32], J2 [25], J3 [26], J4 [38], Co1 [59], Co2 [60], Co3 and McL [24], Fi22

[61], Fi23 [37], Fi240 [47], HS [52], He [13], Ru [62], Suz [63], O0N [64], HN [54], Ly [65], Th [46] and B the baby monster group [66]. The last of these is of particular interest as it was achieved using a 4370-dimensional GF (2)B-module. There are ﬁve sporadic groups not yet covered, M11, M12, M22, M23 and M, the monster group. For the ﬁrst four of these there seems to be no paper giving their maximal subgroups, although the information is well known and can be found easily using modern computational methods. The last, M, is the only sporadic group for which the maximal subgroups are as yet unknown, with the current state of a↵airs described by Wilson [67].

This leaves just the groups of exceptional Lie type to cover, and here progress is more recent. Extensive work by Liebeck and Seitz [44], among others, has yielded a categorisation of the maximal subgroups of the simple exceptional groups of Lie type.

Maximal subgroups for G2(q)wereclassiﬁedin1981byCooperstein[17]forq even and in 1988 by Kleidman [35] for q odd. The only other families of exceptional groups

3 whose maximal subgroups have been classiﬁed are D4(q), also by Kleidman [36], and

F4(q)forqnotapowerof2or3byMagaard[51].Thereareahandfulofcomparatively small exceptional groups outside of these families for which the maximal subgroups have been classiﬁed, the ﬁrst of which is F4(2) by Norton and Wilson in 1989 [55].

Kleidman and Wilson classiﬁed the maximal subgroups of E6(2) in the same year [39] 2 and Wilson published a proof for E6(2) [68], although the classiﬁcation had been CHAPTER 1. INTRODUCTION 11

known for some time. Finally, the maximal subgroups of E7(2) were classiﬁed in 2015 by Ballantyne, Bates and Rowley [8].

This list of finite simple groups whose maximal subgroups have been classified has aparticulargroupwhichisconspicuousbyitsabsence,namelyE8(2). E8(2) is the only exceptional group of Lie type defined over GF (2) without a classification of maximal subgroups, a gap that will hopefully be filled by an as yet unpublished paper of Aubad, Ballantyne, McGaw, Neuhaus, Rowley and Ward [6]. This thesis details the contribution of the author as part of that work to determine non-maximality of certain subgroups of E8(2), as well as determining the conjugacy classes of some of those subgroups. Much of the work done has been computer assisted, making extensive use of the functionality provided by the computer algebra package Magma.

In Chapter 2 we cover the background material required for the work described in this thesis. This starts with a general definition of algebraic groups and some of their properties, including defining the Lie algebra of an algebraic group, maximal tori and through them the root system and associated weights. We also include the Frobenius morphisms necessary for the definition of the Chevalley groups as our context for E8(2).

Chapter 2 continues by exploring work of Liebeck and Seitz in ﬁrst classifying the maximal subgroups of positive dimension in exceptional algebraic groups and then in producing a list of groups in which any maximal subgroup of a simple exceptional group of Lie type must be contained. The chapter ends with a summary of this work in the form of Theorem 2.3, a very important result for all that follows as it guides our analysis and makes a case by case approach to the problem possible.

In the four sections of Chapter 3 we give some preliminary results which are to be of use in this thesis. In Section 3.1 we turn to E8(2) and collect a variety of lists and theorems detailing the information already known about E8(2) which we will use in our analysis. We use Theorem 2.3 to produce an exhaustive list of maximal closed -stable subgroups of positive dimension along with a list, Tables 3.1.1 and 3.1.2, of those almost simple groups whose potential embeddings into E8(2) forms the focus of CHAPTER 1. INTRODUCTION 12 our study. Also included here are several preliminary results that eliminate many of the groups within that list for which there are no embeddings.

We then turn to what is known about the conjugacy classes of E8(2). The centralisers and ﬁxed point dimensions of the involutions and semisimple elements were found by Rowley et al. [7], and we detail those results in Table 3.1.4. This leads to another set of groups from Tables 3.1.1 and 3.1.2 for which it is proven there can be no embeddings of them into E8(2) by consideration of element orders and centraliser sizes. A brief summary of some useful procedures is also included here.

In Section 3.2 we describe some of the theory of modular representations and Brauer characters. This will be of great importance in the calculations detailed in Chapter 6. To complement this we give some definitions of feasible decompositions and fusion patterns as well as much of the Brauer character of E8(2) on its Lie algebra L(G). In Section 3.3 we show how similar calculations can be done for the relevant groups from Tables 3.1.1 and 3.1.2. This requires the theory of dominant weights and is based on work by Steinberg [57] and Lübeck [49]. We also include here the final part of the proof of Theorem 1.1.

Finally, in Section 3.4 we give a method developed by Alistair Litterick and David Craven [48] [18] for determining maximality in groups of Lie type. This method builds from a very important result, Lemma 3.2, by Seitz [56], which allows us to often re- duce the problem to one of ﬁnding ﬁxed vectors of L(G). Chapter 3 ends with a set of lemmas needed for the implementation of the method.

Chapters 4 and 5 contain the main results of the thesis. Chapter 4 details the results of Theorems 1.2 and 1.3 on a case-by-case basis. The methods used in Chapter 4 are mainly those detailed in Chapter 3 and where exceptions occur the method is described in situ. The groups in Theorem 1.4, are the focus of Chapter 5 in which we describe the method and Magma functionality required for their analysis. It then proceeds to detail the results of Theorem 1.4 also on a case by case basis. CHAPTER 1. INTRODUCTION 13

In Chapter 6 we give, for all groups we have encountered, the Brauer character tables along with all feasible decompositions of L(G) and the dimension of H1(S, V ) for their irreducible modules. The tables in this chapter are consistent with the atlas [16] except where noted and have been calculated using the methods from Chapter 3Section3.3.TherearealsotwoappendicescontainingMagma code used for the calculations in this thesis.

Our results can be described by the following four theorems, each summarising a collection of groups from Tables 3.1.1 and 3.1.2 and what is known of their embeddings in E8(2).

Theorem 1.1. Let G = E (2) and H G. Then H cannot be isomorphic to any of ⇠ 8  the following groups:

L2(29) L2(37) L2(41) L2(49) L2(61)

L2(256) L2(512) L2(1024) L3(16) L4(5)

L4(8) U4(8) U3(16) G2(8) F4(4) 2 3 F4(8) F4(8) D4(8) Sp6(8) Sp8(4) + + Sp8(8) ⌦8 (4) ⌦8 (8) ⌦8(8) Sz(32) Sz(128) Sz(512).

Theorem 1.2. Let G = E (2) and H G such that H is maximal. Then H cannot ⇠ 8  be isomorphic to any of the following groups:

L2(7) L2(13) L2(17) L2(27) L4(3)

L3(8) U3(8) G2(3) U3(3) F4(2) 2 3 F4(2)0 D4(2) M11 M12.

Theorem 1.3. Let G = E (2) and H G such that either H = L (3) or H = L (4). ⇠ 8  ⇠ 3 ⇠ 3 Then up to isomorphism the restriction M = L(G) is known and has a direct sum |H decomposition of one of the following forms: CHAPTER 1. INTRODUCTION 14

H = L (3) M = 144 104 ⇠ 3 ⇠ d

H = L (4) M = 20 20 ⇠ 3 ⇠ 4 5 5 5 M = 20 36 ⇠ b 5 5 5 M = 56 . ⇠ b 5 5 5

The modules given here are denoted by their dimension if they are not irreducible and by their name in the relevant character table if they are. Their structure shall be explained in Chapter 4.

Theorem 1.4. Let G ⇠= E8(2) and M either L2(32) or L2(128). Then there is no H G with H maximal such that H = M. Furthermore there are the following upper  ⇠ bounds for the number of conjugacy classes, nc, of subgroups H ⇠= M.

L (32) nc 5, 200, 000 2  L (128) nc 8. 2 

We note that the upper bounds given in Theorem 1.4 are not very close upper bounds; however, it is not unreasonable to hope that future work will improve these signiﬁcantly. We also note that all the results of Theorems 1.1-1.4 concern the simple groups in Tables 3.1.1 and 3.1.2 but take no account of their automorphism groups. This will be dealt with in [6] and so is beyond the scope of this thesis. CHAPTER 1. INTRODUCTION 15 Index of Symbols

Throughout this thesis we use atlas notation for the simple groups with the exception that we use ⌦n±(q)forthesimpleorthogonalgroups.

Symbol Meaning

VX (), Irreducible KX-module of high weight (X reductive) L(X) Lie algebra of X (X an arbitrary algebraic group) V W Direct sum of V and W V + W Module having the same factors as V W X/Y Quotient of X by Y V V ... V Module with socle series V , V /V , ..., V /.../V /V 1| 2| | n 1 2 1 n 2 1 nV V + V + + V (n times) ··· V ⇤ Dual module Hom(V,K) V Restriction of V to the subgroup H |H G Fixed points of in G Z(G) Center of G

p O 0 (G) Minimal normal subgroup of p0 index (G ﬁnite) G0 Connected component of the algebraic group G

F ⇤(H) Fitting subgroup of H GF (q) Finite ﬁeld of q elements

P (i) Projective cover of the module i Chapter 2

Background Material

Before we start looking at E8(2) specifically, we first recall some general theory of ane algebraic groups; for detailed coverage of this material we suggest reference materials of Humphreys [30] and Borovik[9]. For this section K denotes an algebraically closed field of characteristic p 0.

An ane algebraic group over K is a subgroup of GLn(K)forsomeintegern>0 defined by zeros of a collection of finitely many polynomials in the matrix coordinates x , i.e. closed in the Zariski topology. It inherits many properties from GL (K) { ij} n including the Zariski topology and a coordinate ring K[G] as a quotient of K[ x ]by { ij} the ideal of functions vanishing on G. If G is an irreducible variety the we can define the dimension of G to be the transcendence degree of K[G]:G.

We are frequently interested in closed subgroups of an algebraic group G and there are many obvious classes of these [30, 8]. The kernel and image of an algebraic § homomorphism are closed subgroups, where an algebraic homomorphism is a group homomorphism that is also a morphism of varieties. The stabiliser of a vector or sub- space in a rational G-module is a closed subgroup; a rational G-module, V , being a module such that the representation G GL(V ) is an algebraic homomorphism. An ! algebraic group is unipotent if all its elements are unipotent, which is equivalent to it being isomorphic to a group of upper triangular matrices. A semisimple (or reductive) algebraic group is one for which the radical R(G)(orunipotentradicalRu(G)) is trivial. These radicals are the unique largest normal, connected, soluble, (unipotent)

16 CHAPTER 2. BACKGROUND MATERIAL 17 subgroup of G.

Another important property of an algebraic group is its Lie algebra L(G). Many algebraic groups were ﬁrst studied as the automorphism groups of semisimple Lie algebras, see Carter [14]. The study of semisimple algebraic groups and their Lie algebras mirrors that of the semisimple Lie algebras themselves. We can view L(G)asthe space of left-invariant derivations of K[G], which leads to the characterisation of L(G) as the adjoint module, a rational G-module with the same dimension as G. Just as in Carter [14, p.34], this Lie algebra comes equipped with a symmetric bilinear form called the Killing form.

There are two further very important closed subgroups to introduce: a maximal torus T and a corresponding Borel subgroup B. A torus is a connected algebraic group consisting of semisimple elements and it is simultaneously diagonalisable [30, 15] and § therefore isomorphic to a product of the multiplicative group K⇤.Amaximaltorusis atorusthatismaximalinthesetofalltori.IfT is a ﬁxed maximal torus in G then every semisimple element of G is conjugate to an element in T [28], and from this there are several useful consequences. We get that all maximal tori in G are conjugate and have the same dimension, which deﬁnes the rank of G. For a given choice of maximal torus T we also get the Weyl group W = NG(T )/T . A Borel subgroup is a maximal closed, connected, solvable algebraic subgroup of G.

In Carter’s exposition of groups of Lie type [14] the root system is inherited from the Lie algebra of a group. An abstract root system E, for E afinite-dimensional ⇢ Euclidian vector space, is a finite set of vectors satisfying the following four conditions: i) is a spanning set for E, ii) for ↵, ↵ then = 1, iii) for any ↵, the 2 ± 2 reflection ():=–2((↵, )/(↵, ↵))↵ and iv) for any ↵, the number ↵ 2 2 ↵, := 2(↵, )/(↵, ↵) Z. Sometimes condition iv) is not required and a root sys- h i 2 tem that does satisfy it is known as crystallographic. A root system can be defined for any reductive (and hence semisimple) algebraic group in such a way that in con- curs with the root system given by Carter wherever relevant. To do this we need the concept of a weight of a module. CHAPTER 2. BACKGROUND MATERIAL 18

Let G be a reductive algebraic group over K, T amaximaltorusofG and the character group of T ; X(T ) ⇠= Hom(T,K⇤). For any G-module V the restriction of V to T can be decomposed into subspaces called weight spaces. These weight spaces are deﬁned for some X(T )asV := v V : t v = (t)v for all t T .A 2 { 2 · 2 } weight, ,ofV (with respect to T )isanelementofX(T )suchthatV = 0 and the 6 { } multiplicity of is the dimension of V.

We now get our required root system as the non-zero weights of the Lie algebra

L(G). For a root i and its weight space Vi we can deﬁne a root subgroup, Ui ,asthe unique connected unipotent subgroup of G normalised by T such that L(Ui )=Vi . For any maximal torus T and choice of simple roots there is a unique Borel subgroup generated by T and the root subgroups U for +. i i 2

A simple algebraic group is a semisimple algebraic group whose root system is irreducible. The irreducible root systems were classified by Dynkin [21] and when G is semisimple Z(G)isfiniteandG/Z(G) is determined up to isomorphism by the root system of G. There are two (sometimes isomorphic) important semisimple algebraic groups for each irreducible Dynkin diagram, the adjoint and simply-connected groups. These are defined by the character group of their torus X(T ). Given a root system for G we have the Euclidean space E = X(T ) R in which there exists a lattice ⌦Z of weights ⇤ defined as x E such that x, ↵ Z for all ↵ +. We then have 2 h i2 2 Z X(T ) ⇤ and ⇤/Z is finite. The simply-connected and adjoint subgroups are   the extremes of these inclusions, where G is simply-connected when X(T )=⇤ and adjoint when X(T )=Z.

If G is a semisimple algebraic group and ' an algebraic homomorphism from G G ! that is an automorphism of G as an abstract group, then there are two types of pos-

1 sibilities for '. If ' is also an algebraic homomorphism then it is known as an algebraic automorphism. Otherwise by Steinburg [57] the fixed-point subgroup G' is finite and ' is known as a Frobenius morphism. Inner automorphisms are always algebraic automorphisms, as are graph automorphisms, defined as those automorphisms CHAPTER 2. BACKGROUND MATERIAL 19 induced from symmetries of the Dynkin diagram. Turning our attention to Frobenius morphisms, they exist, induced by field automorphisms in positive characteristic p for each power q of p defined by sending K x xk. When is a Frobenius morphism 3 ! the fixed-point subgroup G is a group of Lie type and when G is simple of adjoint

p0 type then O (G) is usually simple. We denote by Aut(G)thegroupgeneratedby the inner automorphisms of G along with graph and ﬁeld morphisms.

2.1 Maximal Subgroups of Algebraic Groups

The study of the subgroup structure of a ﬁnite algebraic group G G splits into  two types; Lie primitive where H G is contained in no proper positive-dimensional  subgroup of G, and non Lie primitive otherwise. Knowledge of the maximal positive dimensional subgroups of arbitrary algebraic groups is therefore useful to the study of ﬁnite subgroups.

A classification of the subgroups of positive dimension started with Dynkin who classified the maximal connected subgroups of a classical group [22]. He also gave a classification of the maximal Lie subalgebras of exceptional complex Lie algebras [23]. This, combined with Humphrey’s result [30, 13] that in characteristic zero there is a § 1-1 correspondence between the Lie subalgebras of L(G)andtheconnectedsubgroups of an algebraic group, gives an equivalent result for exceptional algebraic groups. Since then work by Seitz and Liebeck [56] [42] [43] has extended this classification to positive characteristic and we now have a classification of maximal (not necessarily connected) subgroups of algebraic groups in any characteristic.

Theorem 2.1 (Liebeck Seitz). Let G be an exceptional algebraic group and G G  1  Aut(G). Let M be a proper closed connected subgroup of G which is maximal among proper closed connected NG1 (M)-invarient subgroups of G. Then one of the following holds:

(i) M is either parabolic or of maximal rank;

(ii) G = E , p =2and M =(22 D ).Sym(3); 7 6 ⇥ 4 CHAPTER 2. BACKGROUND MATERIAL 20

(iii) G = E , p =2, 3, 5 and M = A Sym(5); 8 6 1 ⇥ (iv) The connected component of M, M 0, is as in Table 2.1.1 below.

The subgroups M in (ii), (iii) and (iv) exist, and are unique up to conjugacy in Aut(G) and are maximal in G.

Table 2.1.1

G M 0 simple M 0 not simple

G A (p 7) 2 1 F A (p 13), G (p =7), A G (p =2) 4 1 2 1 2 6 E A (p =2, 3), G (p =7), A G 6 2 6 2 6 2 2 C (p =2),F 4 6 4 E A (2 classes,p 17, 19 resp,), A A (p =2, 3), A G (p =2), 7 1 1 1 6 1 2 6 A (p 5) A F , G C 2 1 4 2 3 E A (3 classes,p 23, 29, 31 resp,), A A (p =2, 3), A G G (p =2), 8 1 1 2 6 1 2 2 6 B (p 5) G F 2 2 4

The maximal parabolic subgroups are well known and Liebeck, Seitz and Saxl [41] give the reductive subgroups of maximal rank, ie those that contain a maximal torus. As any Lie imprimitive subgroup is contained in a subgroup of positive dimension we now know all maximal Lie imprimitive subgroups and can turn our attention to Lie primitive groups. For Lie primitive subgroups less is known; however, Liebeck and Seitz [42], extending an earlier result of Borovik [10], give a reduction theorem that limits the possibilities.

Theorem 2.2 (Liebeck Seitz). Let G be an adjoint simple algebraic group of exceptional type over K, algebraically closed of characteristic p>0. Let be a Frobenius morphism of G and G be the ﬁxed point group of G under .LetH be a maximal subgroup of G then one of the following holds:

(i) H = M, where M is a maximal -stable closed subgroup of positive dimension in G;

(ii) H is an exotic local subgroup; CHAPTER 2. BACKGROUND MATERIAL 21

(iii) G = E ,p>5 and H =(Alt(5) Alt(6)).22. 8 ⇥ (iv) H is almost simple;

The groups on part (i) are exactly those found in Theorem 2.1 with G = G 1 h i and those in part (ii) are listed by Liebeck and Seitz [44]. They also give two theorems giving the possibilities for part (iv) for when H Lie(p)[42]andH/Lie(p)[45].We 2 2 can now summarise the results of Section 2.1 with the following theorem:

Theorem 2.3 (Liebeck Seitz). Let H be a maximal subgroup of the ﬁnite exceptional

a group G over Fq, q = p where p is a prime. Then one of the following holds:

(i) H = M where M is a maximal closed -stable subgroup of positive dimension in G; the possibilities are as follows;

(a) Both M and H are parabolic subgroups;

(b) M is a reductive group of maximal rank. The possibilities for M are determined in [41].

(e) M is as in Table 1 of [44], and H = M as in Table 3 of [44].

(ii) H is of the same type as G;

(iii) H is an exotic local subgroup (see [44]);

(iv) G is of type E , p>5 and H (Alt(5) Alt(6)).22; 8 ⇠ ⇥

(v) F ⇤(H)=H0 is simple, and not in Lie(p): the possibilities for H0 are given up to isomorphism by [45];

1 (vi) F ⇤(H)=H(q ) is simple and in Lie(p); moreover rk(H(q )) rk(G), and one 0 0  2 of the following holds:

(a) q 9; 0  2 (b) H(q0) ⇠= A2(16) or A2(16); CHAPTER 2. BACKGROUND MATERIAL 22

(c) q (2,p 1)u(G) and H(q ) = A (q ), 2B (q ) or 2G (q ), where the 0  0 ⇠ 1 0 2 0 2 0 values of u(G) for each type of exceptional group are as follows:

G G2 F4 E6 E7 E8 u(G) 12 68 124 388 1312

In cases (i) (iv), H is determined up to G -conjugacy. Chapter 3

Preliminary Results

3.1 E8(2)

We now turn to E8(2), the focus of this thesis, and outline the information necessary for the methods used in Chapters 4-6. First we use Theorem 2.1 and the results of [41] to determine the maximal subgroups in E8(2) from Theorem 2.3 part (i), which we shall now refer to as maximal restricted subgroups of positive dimension. We then list all the possibilities for F ⇤(H)fromTheorem2.3parts(v)and(vi).Wemakeextensive use of information about the semisimple elements of E8(2), i.e. those of odd order. We therefore include the results of Rowley et al. [7] on the classes of semisimple elements, their centraliser and ﬁxed-point spaces on L(G). The order of E8(2) is as follows

G =2120.313.55.74.112.132.172.19.312.41.43.73.127.151.241.331 | |

3.1.1 Maximal Subgroups

We have the following list of maximal restricted subgroups of positive dimension

(i) Maximal Parabolic Subgroups:

[278]:⌦+ (2) [298] : (Sym(3) L (2)) 14 ⇥ 7 [2106] : (Sym(3) L (2) L (2)) [2104] : (Alt(8) L (2)) ⇥ 3 ⇥ 5 ⇥ 5 [297]:(L (2) ⌦+ (2)) [283] : (Sym(3) E (2)) 3 ⇥ 10 ⇥ 6 92 57 [2 ]:L8(2) [2 ]:E7(2)

23 CHAPTER 3. PRELIMINARY RESULTS 24

(ii) Reductive Subgroups of Maximal Rank:

⌦+ (2) Sym(3) E (2) 16 ⇥ 7 L (2) : 2 3 U (2) : 2 9 · 9 (L (2) E (2)) : 2 3 (U (2) 2 E (2)) : Sym(3) 3 ⇥ 6 · 3 ⇥ 6 2 2 (L5(2)) .4(U5(2)) .4

SU5(4).4 PGU5(4).4 (⌦+(2))2.(Sym(3) 2) ⌦+(4).(Sym(3) 2) 8 ⇥ 8 ⇥ 3 2 3 ( D4(2)) .6 D4(4).6 4 2 4 2 (L3(2)) .GL2(3) [3 ].(U3(2)) .[3 ].GL2(3) 2 (U3(4)) .8 U3(16).8 38.(2.⌦+(2).2) 54.((4 21+4).Alt(6).2) 8 ⇤ 74.(2.(3 U (2))) 112.(5 SL (5)) ⇥ 4 ⇥ 2 132.(12 GL (3)) 312.(5 SL (5)) ⇤ 2 ⇥ 2 151.30 331.30

We now need a list of possibilities for F ⇤(H)givenbyTheorem2.3parts(v)and (vi)

Alt(5 17) L (7) L (11) L (13) L (17) 2 2 2 2 L2(19) L2(25) L2(27) L2(29) L2(31)

L2(37) L2(41) L2(61) L3(3) L3(5)

L4(3) L4(5) PSp4(5) U3(3) G2(3)

M11 M12 J2 J3

Table 3.1.1: Simple groups / Lie(2) 2 CHAPTER 3. PRELIMINARY RESULTS 25

L2(8) L2(16) L2(32) L2(64) L2(128) L2(256) L2(512)

L2(1024) L3(4) L3(8) L3(16) L4(4) L4(8) L5(2)

L5(4) L5(8) U3(4) U3(8) U3(16) U4(2) U4(4)

U4(8) U5(2) U5(4) U5(8) Sp4(4) Sp4(8) Sp6(2)

Sp6(4) Sp6(8) Sp8(2) Sp8(4) Sp8(8) Sz(8) Sz(32) + + + Sz(128) Sz(512) ⌦8 (2) ⌦8 (4) ⌦8 (8) ⌦8(2) ⌦8(4) 3 3 3 ⌦8(8) D4(2) D4(4) D4(8) F4(2) F4(4) F4(8) 2 2 F4(2)0 F4(8)

Table 3.1.2: Simple groups Lie(2) 2

The groups missing from Table 3.1.2 are those isomorphic to some group in Table

3.1.1, i.e. L2(4) ⇠= Alt5, L3(2) ⇠= L2(7) and L4(2) ⇠= Alt8.

Using Lagrange’s theorem we can immediately remove the following groups from consideration.

F ⇤(H) F ⇤(H) Invalid divisor | | 2 L2(29) 2 .3.5.7.29 29 2 2 L2(37) 2 .3 .19.37 37 2 L2(61) 2 .3.5.31.61 61 7 2 6 6 L4(5) 2 .3 .5 .13.31 5 32 5 4 2 Sp8(4) 2 .3 .5 .7.13.17 .257 257 24 4 3 ⌦8(4) 2 .3 .5 .7.13.17.257 257 3 36 6 2 2 2 D4(8) 2 , 3 .7 .19 .37.73 .109 109 48 6 4 2 2 2 F4(4) 2 .3 .5 .7 .13 .17 .241.257 257 72 10 2 4 2 2 2 F4(8) 2 .3 .5 .7 .13 .17.19 .37.73 .109.241 109 2 36 5 2 2 2 F4(8) 2 .3 .5 .7 .13 .19.37.109 109 8 L2(256) 2 .3.5.17.257 257 Sz(128) 214.5.29.113.127 113 Sz(512) 218.5.7.13.37.73.109 37

Table 3.1.3: Invalid group orders CHAPTER 3. PRELIMINARY RESULTS 26

3.1.2 Elements of E8(2)

We now wish to detail various properties of the elements in E8(2) up to conjugacy. We start with the elements of order 2, of which there are four classes.

Proposition 3.1. Let G ⇠= E8(2), V = L(G)andt be an involution in G. Also let

U = Ru(CG(t)) and L be such that CG(t)=UL. Then the possibilities for t are as follows:

(i) If t 2A, then dim(C (t)) = 190, U = 21+56 and L = E (2), 2 V ⇠ ⇠ 7 (ii) If t 2B, then dim(C (t)) = 156, U [278]andL = Sp (2), 2 V ⇠ ⇠ 12 (iii) If t 2C, then dim(C (t)) = 138, U [281]andL = Sym(3) F (2), 2 V ⇠ ⇠ ⇥ 4 (iv) If t 2D, then dim(C (t)) = 128, U [284]andL = Sp (2). 2 V ⇠ ⇠ 8

Proof. For the shape of CG(t) see Aschbacher, Seitz [4]. The dimension of CV (t)can be calculated directly in Magma.

We now consider the semisimple elements of E8(2). Table 3.1.4 contains the results of [7] in which the semisimple conjugacy classes of E8(2) are identified with their corresponding number in Lübeck’s ordering from [50] where much of the data was determined. It also contains the shape and size of the centraliser of the element, as well as its fixed space dimension and the class of its powers. CHAPTER 3. PRELIMINARY RESULTS 27 ------3C 3C 3C 3C 3B,5B 3B,5A 3B,5A 3C,5A 3D,5B 3A,5A 3D,5A Powers )) x ( V 92 86 80 68 48 80 38 48 34 30 28 28 32 20 48 34 28 26 24 20 16 32 16 14 C 248 134 dim( 127 · 73 · 43 · 19 73 43 43 31 · · · · (2). 31 · 8 · 17 41 31 19 31 E 17 17 31 · · 2 · · · 17 · · · 2 17 2 · 2 17 2 13 · 7 17 13 13 11 17 17 19 13 17 11 19 13 13 11 17 · · 11 · 3 · · · · · · 13 7 · · · · · · 13 · · · 17 | · | 5 19 7 2 · · 4 2 2 ) · · 3 2 2 5 7 · 7 7 · · 5 11 7 7 5 5 2 13 2 x 13 11 5 5 7 17 · · 11 · · 5 · 6 · 4 · · · · ( (2) 5 · 5 5 · · · · · · 11 · · 8 7 3 8 2 3 3 6 · 2 G · · 5 7 5 6 5 4 2 2 · 4 7 3 3 · 5 5 · 3 7 7 3 3 3 3 3 E 5 6 4 7 3 3 3 C 3 · 13 · · 2 | · · · · 8 · | · 3 · · · · · 3 3 · · · · · · 7 2 2 8 6 2 9 4 2 4 · · 2 2 11 10 6 6 4 · 5 3 3 15 13 10 10 2 5 5 2 12 3 5 2 2 2 2 2 · 3 · · 2 2 2 2 11 · · 13 12 2 · · 5 · 2 2 6 12 6 · 20 20 13 3 3 20 3 3 2 2 3 7 · 2 · · 10 · · 3 30 2 36 36 · 39 2 5 2 2 2 · 42 2 12 3 · 63 2 3 . (2) (2) (8) (2) 4 (2) (2) (2) 5 3 4 4 4 (2) 8 D (2)) U U L 3 U 3 D 3 ⌦ (2) (2) 3 (2) (2) (4) (2) (4) (2) (2) (2) ) 4 ⇥ ⇥ ⇥ U (2) (2) (16) (16) (2) ⇥ 5 3 5 3 ⇥ 8 + 10 7 6 x 2 2 9 ⇥ (4) 14 12 10 ⇥ D ( U U U U (2) 5 ⇥ ⌦ ⌦ U L L 3 E E PGU 8 ⌦ ⌦ ⌦ G (2) · (4) :2 3 ⇥ ⇥ ⇥ ⇥ 2 ⇥ E 3 ⇥ (2) ⇥ ⇥ ⇥ C ⇥ ⇥ ⇥ SU ⇥ ⇥ ⇥ (2) 2 3 L 3 6 3 7 3 ⇥ 3 5 9 11 13 15 15 L 17 GU 15 Sym(3) Sym(3) 15 17 13 Sym(3) E ⇥ ⇥ 2 ⇥ 19 ⇥ ⇥ ⇥ ( ⇥ . 5 15 7 5 9 9 3 9 Table 3.1.4: Conjugacy classes of semisimple elements of 1 294 376 147 258 480 247 441 516 560 656 580 366 679 712 709 540 636 686 621 600 706 695 738 693 823 ¨bc Number Lübeck A A A A F C G E B B D A A A A A C C B B B B D D AB CD 1 3 5 7 9 3 9 3 5 7 9 3 9 11 13 15 19 15 15 15 15 13 15 15 17 17 Conjugacy Class CHAPTER 3. PRELIMINARY RESULTS 28 - - - - 3B,7B 3C,7B 3A,7B 3B,7A 3A,7B 3C,7A 3D,7B 3A,7A 5A,7A Powers 3B,13B 3B,11A 3C,11A 3C,13A 3A,11A 3A,13A 3D,11A 3B,17AB 3C,17AB 3A,17AB 3D,17CD 3C,9C,5A,15D 3C,9A,5A,15D 3C,9A,5A,15D )) x ( V 8 8 8 8 38 32 32 26 20 20 14 14 28 20 16 14 12 20 14 14 10 20 12 10 20 14 14 C dim( 2 31 17 17 31 2 · · 7 7 13 · · 7 11 13 17 · · 3 2 2 2 · 3 2 7 | · · · · 7 7 7 5 5 11 11 11 13 7 17 7 7 7 2 2 3 ) 43 · 13 17 · · · · · · · · · · · · · · 5 5 5 5 7 5 7 · x 2 5 · · 41 · · · 4 5 4 4 4 · · · 5 5 4 4 4 · ( 5 5 5 5 2 3 · 3 3 3 3 3 3 3 3 3 5 3 5 6 4 4 · G 5 4 3 · · · 5 31 3 · · · · · · · · · · · · · 3 3 3 3 3 5 3 3 · 3 5 5 2 C 3 · · · 6 6 3 2 | · · · 4 3 2 3 3 · 2 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 9 6 6 · 6 4 3 · · · 12 2 2 2 2 2 2 6 2 15 12 10 2 2 2 2 2 (8) 2 (8) (4)) : 3 : 2Alt(4) 2 3 L Alt(5) (8) : 2Alt(4) L L 2 (2) (2) (2) ⇥ (2) (2) (2) (2) (2) (2) 3 ⇥ ) 1+2 + 4 L ⇥ Alt(5) 6 5 4 4 4 4 8 ⇥ 3 Sym(3) x : 2Alt(4) : 2Alt(4) : 2Alt(4) 2 1+2 + Sym(3) D ( U U L L L L 8 ⌦ ⇥ Sym(3) 3 3 ⇥ GU ⇥ G ⇥ 31 195 205 255 (2) ⇥ 3 Q ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ 3 3 ⇥ ⇥ ⇥ 1+2 + 1+2 + 1+2 + C 3 ⇥ ⇥ : 3 3 3 3 L ⇥ ⇥ 2 Sym(3) 33 35 21 31 45 51 ⇥ 21 Sym(3) ⇥ 21 17 (2) ⇥ ⇥ ⇥ ⇥ 129 (3 21 3 ⇥ 45 . ⇥ 45 33 3 L Sym(3) 21 33 13 21 13 51 ⇥ ⇥ ⇥ 7 7 11 610 720 728 469 594 697 760 826 672 857 768 748 811 790 778 762 820 872 864 837 773 798 853 783 764 832 870 ¨bc Number Lübeck A A A A F F C G C C E E B B B D D H AB AB AB EF CD CD GH ABC ABC 21 35 39 45 21 33 21 21 39 45 21 33 21 39 45 21 31 21 33 41 51 51 33 51 51 31 43 Conjugacy Class CHAPTER 3. PRELIMINARY RESULTS 29 - - Powers 5B,13B 5A,13B 7B,13A 3C,19A 5A,11A 3A,19A 7A,13A 3D,19A 5B,17CD 5A,17AB 7A,17AB 3B,31ABC 3A,31ABC 3C,9B,7B,21F 3C,9D,7B,21F 3C,9B,7A,21D 3C,9A,7A,21D 3C,9B,13A,39B 3C,9C,11A,33CD 3C,9A,11A,33CD 3B,5A,7A,15A,21C,35A 3A,5A,7A,15B,21A,35A 3C,5A,7A,15D,21D,35A )) x ( V 8 8 8 8 8 8 8 8 14 16 12 10 10 12 14 12 14 14 12 10 14 12 10 10 10 C dim( 7 7 7 7 13 17 13 31 31 2 73 · · · 2 | · · · · · · 7 7 19 · 7 ) 11 13 5 5 5 11 17 17 · · · 2 2 5 5 7 5 127 · · · x 19 19 13 7 13 11 · · · · · · 5 7 · 4 4 · · · 4 ( · 3 · · · · · · 3 3 3 4 3 · · 3 3 5 3 5 7 3 G 2 2 2 3 3 3 3 3 2 2 2 3 3 3 3 3 · · · · · · 3 3 3 3 3 3 · · · · 3 3 5 7 3 C · · · · · 4 | · · · · · 3 2 3 3 3 2 2 2 2 2 3 2 3 2 2 2 2 3 3 2 2 2 2 2 2 2 2 2 2 2 (2) 3 2 2 (2) L 3 9 (2) 2 ⇥ (2) (2) (2) 3 ) 3 7 3 ⇥ 5 Sym(3) 3 3 3 Sym(3) Sym(3) x ( L L L ⇥ ⇥ ⇥ Alt(5) Alt(5) Alt(5) Sym(3) Sym(3) ⇥ ⇥ Sym(3) GU PGU Sym(3) 19 G ⇥ ⇥ Sym(3) · 165 255 357 ⇥ ⇥ ⇥ 3 3 ⇥ ⇥ ⇥ ⇥ ⇥ C 7 3 57 91 99 ⇥ ⇥ ⇥ ⇥ 13 ⇥ ⇥ ⇥ 3 Sym(3) 73 91 93 ⇥ ⇥ 63 65 93 35 99 85 117 127 105 ⇥ 19 63 63 105 63 877 823 861 863 754 802 843 849 800 858 814 804 870 817 865 808 788 841 867 829 794 851 845 878 835 ¨bc Number Lübeck C D A C E D D AB AB AB AB AB EF CD DE ABC ABC ABC ABC DEF FGH 55 57 63 63 91 ABCD ABCD CDEF 105 105 57 85 99 65 99 57 105 119 63 91 93 93 63 117 65 73 85 ABCDEF GHI Conjugacy Class 127 CHAPTER 3. PRELIMINARY RESULTS 30 - - - Powers 5B,41AB 3A,43ABC 5A,31ABC 7A,31ABC 3D,43ABC 3B, 43ABC 3A,73ABCD 7A,73ABCD 3C,9D,19A,57AB 3C,9A,17AB,51EF 3A,127ABCDEFGHI 3D,5A,11A,15C,33AB,55A 3A,7B,13A,21B,39A,91ABC 3B,5B,13B,15F,39C,65ABCD 3B,5A,15A,17AB,51AB,85AB 3A,5A,15B,17AB,51CD,85AB 3B,7A,17AB,21C,51AB,119AB 3D,5B,15G,17CD,51GH,85CDEF 3B,5A,15A,31ABC,93DEF,155ABC 3A,7A,21A,31ABC,93ABC,217ABCDEF 3C,5A,7A,9A,15D,21D,35A,45A,63D,105AB )) x ( V 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 10 10 10 10 C dim( 17 31 | 7 · ) 43 19 17 · 31 11 13 17 13 17 31 31 · · · · x 43 43 17 5 · · · · · · · · 41 7 73 73 ( 5 127 · · · · 2 3 5 · · · · 5 5 5 5 7 7 5 7 · G · 3 3 151 241 331 · 2 2 2 2 · · · · · · · · 5 3 3 7 2 · · 3 3 3 3 C 3 2 | · 3 3 3 3 3 3 3 3 3 · 2 2 3 2 2 3 3 3 ) x ⇥ ⇥ ⇥ ( Sym(3) Sym(3) Sym(3) Sym(3) G 151 153 465 165 195 205 219 241 255 273 315 331 357 381 465 511 651 ⇥ ⇥ ⇥ ⇥ C 129 129 255 129 171 217 255 837 859 859 868 879 876 877 847 872 864 839 875 866 855 860 870 873 871 869 878 874 876 862 880 ¨bc Number Lübeck AB AB AB EF GHI ABC ABC ABCD ABCD ABCD ABCD ABCDE 153 165 315 255 ABCDEF ABCDEF ABCDEF ABCDEF ABCDEF JKLMNO 129 155 273 195 219 255 357 GHIJKLMN ABCDEF GH ABCDEF GH 151 ABCDEF GHI ABCDEF GHIJ 129 171 217 465 651 129 ABCDEF GHIJK Conjugacy Class 255 205 511 381 241 331 CHAPTER 3. PRELIMINARY RESULTS 31

One immediate consequence of the information in Table 3.1.4 is knowledge of the exponents of the Sylowp subgroups of G. The exponent of a Sylow3 subgroup is 9 and all other Sylowp subgroups are elementary abelian. This allows us to rule out another set of subgroups whose Sylowp subgroups are incompatible with those of G.

F ⇤(H) p prime Invalid Exponent of Sylowp

L2(49) 7 49

L2(512) 5 25

L2(1024) 3 27 Sz(32) 5 25

Sp6(8) 3 27

Sp8(8) 3 27 + ⌦8 (8) 3 27

⌦8(8) 3 27

Table 3.1.5: Invalid Sylow Exponents

Finally, we can also rule out L4(8) and U4(8) by the order of the centraliser of an element of order 13. In L (8) there exists x of order 13 such that C (x) =32.5.13 4 | L4(8) | which does not divide either centraliser order in E8(2); likewise, in U4(8) there exists x of order 13 such that C (x) =5.7.13 which does not divide either centraliser | U4(8) | order in E8(2).

In the course of this thesis we often need to ﬁnd the centraliser of a semisimple

2 t 1 element g or all inverting involutions for it, i.e. all t G such that t =1andg = g . 2 In smaller groups than E8(2) this is a fairly straightforward calculation; however, due to the size of the group normal methods do not work here. We therefore introduce methods for ﬁnding these two objects developed by Rowley and Ballantyne for use in [6].

Given a semisimple element x G = E (2) and a centraliser C (y)wherey xG, 2 ⇠ 8 G 2 the code in Appendix B returns a subgroup of CG(x)thathasaverygoodchanceof being the whole centraliser. As we know the orders of all semisimple centralisers we CHAPTER 3. PRELIMINARY RESULTS 32 can easily check whether we have indeed found the whole centraliser.

For inverting involutions suppose g G = E (2) is an element of odd order, then 2 ⇠ 8 all inverting involutions are contained in the subgroup

x x 1 C (g),t = C⇤ (g)= x G g = g or g = g . h G i G { 2 | }

We therefore only need to find a single inverting involution and CG(g)usingthecen- traliser code described earlier. To find an inverting involution for g we use the following procedure. First we find an involution t and y G such that h = tty gG. Then 2 2 t inverts h and we can use them as a guide for where to search for an inverting involution for g. We can calculate K = CG(h)anduseBray’smethodtofindCK (t).

Choosing an element z of odd order out of CK (t)insuchawaythatanequivalent element of CG(g)canbechosen,wenowknowthatCG(z)containsaninvertinginvo- lution for g. In some cases more must be done to ﬁnd an inverting involution inside

CG(z). For our purposes this is easy, as we can employ LMGRadicalQuotient to get

CG(z)=CG(z)/ z in which the inverse image of N ( g )containsaninverting h i CG(z) h i involution, and is very small. Examples of the odd order element z that are picked are as follows

g 3C z C (g), z 13A 2 2 G 2 g 3B z C (g), z 11A, 13B, or 17AB 2 2 G 2 g 3D z Syl (C (g)), z 7B 2 h i2 7 G 2 g 7B z C (g), z 13A 2 2 G 2 g 31ABC z C (g), z 5A 2 2 G 2

3.2 Brauer Characters

An important tool in studying embeddings of groups is that of the Brauer characters. Just as ordinary characters encode the information about the eigenvalues of an element of a complex representation, so too do the Brauer characters on a modular representation. For a GF (q)G-module V with q = pn we can deﬁne a complex valued function V on the set of p-regular conjugacy classes in the following way. For ordinary representations the value of 'V (x)isthesumoftheeigenvaluesoftheactionofx on CHAPTER 3. PRELIMINARY RESULTS 33

V ; however, in finite fields we lose information when taking such a sum so we first map the values into a field of characteristic 0, i.e. C. This map, known as a lifting, th takes the elements of the multiplicative group GF (q)⇤ and maps them to the q roots of unity. We can now sum the images of our eigenvalues and we have our required complex valued function.

An important fact of modular representation theory is that Maschke’s theorem frequently doesn’t hold as p - G is a requirement for it to do so. Whilst this makes | | ﬁnding the set of indecomposable modules for G a much harder task it also allows for deeper insight if they are well understood. In Section 3.2.3 we use the irreducible Brauer characters of H G to determine the feasible decompositions of the restriction  L(G) . |H

3.2.1 Conway Polynomials

In order for the Brauer characters calculated for H G to be consistent with the  restrictions of those of G we must fix our lifting of the finite fields. We do this in the way introduced in the atlas [33] via the Conway polynomials. Up to isomorphism there is a unique field GF (q)oforderq = pn, p prime n N.Forn =1,GF (p) 2 is the field Z/pZ and then GF (q)maybedefinedasGF (p)[X]/(fn)wherefn is an irreducible polynomial of degree n. For our consistent lifting to be possible, we must

th ﬁx an fn for each n which will be called the n Conway polynomial.

This choice is made with certain conditions in mind; fn is a monic and primitive polynomial ensuring that zn = X +(fn) is a generator for the multiplicative group

GF (q)⇤. We also require fn to be consistent with fd where d is any divisor of n, i.e. if ↵ =(pn 1)/(pd 1) then z↵ is a root of f . A lexographic ordering is now deﬁned n d on the set of all such polynomials as

n n 1 n n n 1 n X an 1X + +( 1) a0

Having now got an fixed definition for GF (q)andageneratorforitsmultiplicative group z = X +(f )weusetheliftingz ⇣ =exp(2⇡i/(q 1)). We note that n n n ! in Magma this choice of finite field definition and lifting is automatic wherever its database knows the nth Conway polynomial, which in characteristic 2 is for n 94. 

3.2.2 Brauer Character of L(G)

E8(2)-class Brauer character value E8(2)-class Brauer character value 3A 77 21E 0 3B 14 21F 12 3C 5 21G 0 3D -4 21H 3 5A 23 31ABC 23+c31+6&5 ( 3, 5) 5B -2 31D 0 ⇤ ⇤ 7A 52 33AB 21+6b33 ( ) 7B 3 33CD 12 +3b33 (⇤⇤ ) 9A 29 33E 0 ⇤⇤ 9B 8 33F 3 9C 2 35A 2 9D -1 39A -1 11A 6 39B 5 13A 14 39C 1 13B 1 41AB 2 15A 44 43ABC 4-c43 ( 3, 7) 15B 2 45A 14⇤ ⇤ 15C 11 45B -1 15D 5 45C 2 15E -1 51AB 12+v51&2&19&35 ( 5) 15F 4 51CD v51&2&19&35 ( 5)⇤ 15G 1 51EF 4+2v51&2&19&35⇤ ( 5) 17AB 22+b17 ( ) 51GH 4+v51&2&19&35 ( ⇤5) 17CD 2+b17 ( ⇤⇤) 55A 1 ⇤ 21A 7 ⇤⇤ 57AB 16+3b57 ( ) 21B 21 57C 1 ⇤⇤ 21C 7 57DE 6+b57 ( ) 21D -2 ... ⇤⇤

Table 3.2.1: Brauer Values of semisimple classes of E8(2) on L(G)

Table 3.2.1 contains the Brauer character values of the conjugacy classes of E8(2) on L(G). These values were calculated either using the code in Appendix B or by CHAPTER 3. PRELIMINARY RESULTS 35 hand. The algebraic irrationalities, both here and in Chapter 6 were converted into atlas notation, described in [16], by use of the method outlined by Bosma [11].

3.2.3 Feasible Decompositions

Deﬁnition 3.2. A Fusion pattern from H to G is a map f from the p-regular conjugacy classes of H to the conjugacy classes of G which preserves element orders.

This deﬁnition is a weaker version of the deﬁnition given by Frey [27] and Litterick [48] as it doesn’t take account of power maps.

Deﬁnition 3.3. A feasible decomposition of H on a ﬁnite-dimensional G-module V is a KH-module V0 such that, for some fusion pattern f, the Brauer character at x for any x H on V is equal to the Brauer character at x on V . The Brauer character 2 0 of V0 is then called a feasible character.

This deﬁnition is again a weaker version of that given by Litterick [48], and, like him, we note that not every feasible decomposition or fusion pattern corresponds to an embedding of H in G.

For a given group from Tables 3.1.1 or 3.1.2 we wish to determine all feasible decompositions of L(E8(2)). To do so we need the Brauer character values of all irreducible GF (2)H-modules of dimension 248. These are calculated either directly  in Magma or by using the theory in Section 3.3. We also need the Brauer character values of E (2) on L(G)whicharegiveninTable3.2.1forelementsoforder 57 8  and can be calculated using the code in Appendix B. Whilst it is possible to calculate the feasible decompositions by solving a set of simultaneous equations, we instead approach this problem by e↵ectively constructing all decompositions of 248-dimensional GF (2)H-modules and checking which are feasible decompositions using the code in Appendix A. The results of this analysis for the relevant groups from Tables 3.1.1 and 3.1.2 are presented in Chapter 6.

Lemma 3.1. Let G = E (2) and H G, then H is not isomorphic to L (41), L (16) ⇠ 8  2 3 or G2(8).

Proof. In Chapter 6 L2(41), L3(16) and G2(8) are found to have no feasible decompositions and therefore there are no embeddings of these groups into E8(2). CHAPTER 3. PRELIMINARY RESULTS 36 3.3 Irreducible Modules of Groups of Lie Type

Whilst there are algorithms in Magma for calculating the irreducible modules of a group G, these sometimes are insucient and we must use the theory of highest weights to ﬁnd the irreducible modules of dimension 248. 

For the rest of this section we let G be a connected reductive simple algebraic group of simply-connected type and rank r over K an algebraically closed ﬁeld of characteristic p>0. Let be a Frobenius endomorphism of G and G(q)=G be the ﬁxed point subgroup of . We denote by G˜ the quotient G/Z(G)andusethenotation from Chapter 2 for tori and weights.

A weight ! ⇤ is called dominant if !, ↵ 0forall↵ +. For a choice of sim- 2 h i 2 ple roots ↵ ,...,↵ X(T )ofG with respect to T we can deﬁne a set, ! ,...,! , { 1 r} ⇢ { 1 r} of fundamental dominant weights as the weights such that ! , ↵ = . Any domi- h i ji ij nant weight can be written as a non-negative linear combination of the fundamental dominant weights. The Weyl group W of G acts on X(T )andunderthisactioneach W -orbit on X contains a unique dominant weight. There is then a partial ordering of

X(T ) deﬁned by ! !0 if, and only if, !0 ! is a non-negative linear combination of  the simple roots.

Let V be a ﬁnite-dimensional G-module over K, then considering V as a T -module there is a direct sum decomposition V = ! X(T ) V! into weight spaces V! such that 2 t T acts by multiplication with !(t)onLV!. A theorem of Chevalley characterises 2 the irreducible modules of G via their set of weights.

Theorem 3.4 (Chevalley). Let G and V be as above.

(i) If V is irreducible then the set of weights of V contains a (unique) element such that for all weights ! of V we have ! . This is called the highest  weight of V , it is dominant and we have dim(V)=1.

(ii) An irreducible G-module V is determined up to isomorphism by its highest weight.

(iii) For each dominant weight X(T ) there is an irreducible G-module V () with 2 CHAPTER 3. PRELIMINARY RESULTS 37

highest weight .

Proof. See Humphreys [31]

A dominant weight = a ! + + a ! X(T )iscalledp-restricted if 0 a 1 1 ··· r r 2  i  p 1for1 i r. Steinberg’s tensor product theorem gives a way of constructing   all highest weight modules V ()ofg from those with p-restricted highest weights.

Theorem 3.5 (Steinberg). Let F0 be the Frobenius automorphism of K of characteristic p, raising elements to their pth power. Twisting the G-action on a G-module V

i (i) with F0,i Z 0, we get another G-module which we denote by V .If0,...,n are 2 p-restricted weights then

V ( + p + + pn ) = V ( ) V ( )(1) V ( )(n). 0 1 ··· n ⇠ 0 ⌦ 1 ⌦ ···⌦ n

Proof. See Humphreys [31]

This pairs with another theorem of Steinberg to give us the result we want linking the irreducible modules of G with those of G(q).

Theorem 3.6 (Steinberg). Let G and G(q) be as before. We deﬁne a subset ⇤ of dominant weights. If G(q) is not of type 2B , 2G or 2F then ⇤ = a ! + +a ! 0 2 2 4 { 1 1 ··· r r|  a q 1 for 1 i r .IfG(q) is of type 2B , 2G or 2F then ⇤ = a ! + + i    } 2 2 4 { 1 1 ··· ar!r 0 ai (q/pp) 1 if ↵i is a long root, 0 ai (qpp) 1 if ↵i is a short root |     } (note that q is the square root of an odd power of p=2,3 or 2, respectively, in these cases). Then the restictions of the G-modules V () with ⇤ to G(q) form a set of pairwise 2 inequivalent representatives of all equivalence classes of irreducible KG(q)-modules.

Proof. See Humphreys [31]

These three theorems show that from knowledge of the dimensions of the modules V () we can get the dimensions of all irreducible modules of G(q). We note that the dimension of V ()=V ()(i). It is also worth noting that with some further e↵ort it would also be possible to determine all irreducible modules for G˜, however for our purposes G(q)=G˜ in all cases. CHAPTER 3. PRELIMINARY RESULTS 38

3.3.1 Dimensions of Highest Weight Modules

L¨ubeck has determined the dimension of the highest weight module for all simple types over all characteristics with some upper limits on dimensions [49]. For our purposes all highest weight modules are of small enough dimension to be included, those that are too high are marked N/A but have dimension higher that 248 so are not needed. We will need the following highest weight dimensions, all of which are for characteristic 2

(as Bi ⇠= Ci in characteristic 2 we show only Ci for convenience):

Dimensions

Highest Weight A2 C2 [0, 0] 11 [1, 0] 34 [0, 1] 34 [1, 1] 816

Table 3.3.1: Rank 2 groups

Dimensions

Highest Weight A3 C3 [0, 0, 0] 11 [1, 0, 0] 46 [0, 0, 1] 48 [0, 1, 0] 614 [1, 0, 1] 14 48 [0, 1, 1] 20 64 [1, 1, 0] 20 112 [1, 1, 1] 64 512

Table 3.3.2: Rank 3 groups CHAPTER 3. PRELIMINARY RESULTS 39

Dimensions

Highest Weight A4 C4 D4 [0, 0, 0, 0] 111 [0, 0, 0, 1] 588 [1, 0, 0, 0] 5168 [0, 0, 1, 0] 10 26 26 [0, 1, 0, 0] 10 48 8 [1, 0, 0, 1] 24 128 48 [0, 0, 1, 1] 40 160 160 [0, 1, 0, 1] 40 246 48 [1, 0, 1, 0] 40 416 160 [1, 1, 0, 0] 40 768 48 [0, 1, 1, 0] 74 784 160 [1, 0, 1, 1] 160 2560 784 [1, 1, 0, 1] 160 3696 246 [0, 1, 1, 1] 280 N/A 784 [1, 1, 1, 0] 280 N/A 784 [1, 1, 1, 1] 1024 N/A 4096

Table 3.3.3: Rank 4 groups

3.4 Determining Maximality

Having now found the feasible characters for the groups we are interested in we now turn to determining if they are maximal in E8(2). To this end we shall now outline some methods developed by Litterick [48] and Craven [18]. The basis of this method relies on the following Lemma from Seitz [56]:

Lemma 3.2 (Seitz). Let 0 = v L(G) and C = C (v). Then 6 2 G (i) If v is semisimple then C contains a maximal torus of G. (ii) If v is nilpotent then R (C) =1and hence C is contained in a proper parabolic u 6 subgroup of G.

To make use of this lemma we must observe the following from the theory of algebraic groups. Let G be an algebraic group over an algebraically closed field K, G be CHAPTER 3. PRELIMINARY RESULTS 40 the fixed point group under some Frobenius morphism of G and H G .Weknow  if H fixes a nonzero vector on L(G)thenH fixes the same nonzero vector v on L(G). Then H is contained in the closed stabiliser subgroup of v, C (v)= g G g.v = v . G { 2 | } We also use the Jordan decomposition of L(G), which states for w L(G)thereexists 2 auniquesumws +wn = w where ws is semisimple and wu is nilpotent. As this decomposition is unique, any endomorphism of L(G)fixingw also fixes wu and ws, thereby implying that if H fixes a non-zero vector on L(G)italsofixeseitherasemisimpleor nilpotent vector so Lemma 3.2 applies.

We are therefore interested in conditions that, for a given feasible decomposition, will imply a ﬁxed vector in L(G). To do this we need to use the theory of cohomology groups. For a group S and an KS-module V the cohomology group H1(S, V )isthe quotient of the additive group Z1(S, V )of1-cocyclesbythesubgroupB1(S, V )of 1-coboundaries. Whilst this deﬁnition is fairly technical for our purposes it suces that the cohomology group parameterises the classes of non-split exact sequences of KS-modules 0 V E K 0 { } ! ! ! ! { } up to equivalence under isomorphisms inducing the identity on V and K.Fromthis and Jansen [34] we get to

1 1 1 Ext (V,W) = Ext (K, V ⇤ W ) = H (S, V ⇤ W ) S ⇠ S ⌦ ⇠ ⌦

1 where V and W are KS-modules and ExtS(V,W) is the set of equivalence classes of short exact sequences of S-modules:

0 W E V 0. ! ! ! ! The following result of Litterick [48] is often very useful in determining ﬁxed vectors, whose proof we shall include as it is instructive for what follows.

Lemma 3.3 (Litterick). Let S be a ﬁnite group and M a ﬁnite-dimensional KS- module, with composition factors W1,...,Wr, of which m are trivial. Set n = dim H1(S, W ), and assume H1(S, K)= 0 . i { } P (i) If n

(ii) If m = n and M contains no nonzero trivial submodule, then H1(S, M)= 0 , { }

(iii) Suppose that m = n>0, and that for each i we have H1(S, W )= 0 i { } () 1 H (S, W ⇤)= 0 . Then M has a nonzero trivial submodule or quotient. i { } Proof. In each case, we proceed by induction on the number r of composition factors of M.

(i) If r =1orifM has only trivial composition factors, the result is immediate. So let r>1andassumethatM has a nontrivial composition factor. Let W M be ✓ asubmodulewhichismaximalsuchthatM/W has a nontrivial composition factor.

1 Let n0 = dim H (S, Wi), the sum being over composition factors of W , and let m0 be theP number of trivial composition factors of W . If n n0

(ii) Suppose that n = m,thatM has no trivial submodule and that H1(S, M) =0, 6 so that there exists a non-split extension 0 M N K 0 . Since M con- { } ! ! ! ! { } tains no trivial submodules, neither does N. However, this contradicts (i), since N has

1 m +1trivialcompositionfactors,whilethesum dim H (S, Wi)overcomposition factors Wi of N is equal to m. P

(iii) Now suppose n = m>0. Assume that M has no trivial submodules. We will show that M has a trivial quotient. Let N be a maximal submodule of M. Then N has no nonzero trivial submodules (since M doesn’t). Hence H1(S, M/N)= 0 , { } otherwise N would have a trivial submodule by part (i), a contradiction. By induction on r, we deduce that N has a trivial quotient. Let Q be a maximal submodule of N such that N/Q is trivial. Then M/Q is an extension of a trivial module by the irreducible module M/N. By our hypothesis on cohomology groups, we have

1 1 1 Ext (M/N,K) = Ext (K, (M/N)⇤) = H (S, (M/N)⇤)= 0 . Thus the extension S ⇠ S ⇠ { } splits and M has a trivial quotient, as required. CHAPTER 3. PRELIMINARY RESULTS 42

Lemma 3.3 is very powerful and allows for many feasible decompositions to be ruled out from leading to maximal subgroups; however, when it fails to do so we must turn to more computationally intense methods. For this we must use the theory of projective covers, which we shall now brieﬂy outline from Alperin [1].

For a KS-module V we deﬁne several important submodules or quotients. The Ja- cobson Radical, Rad(V ), is the intersection of all maximal submodules and V/Rad(V ) known as the head of V is necessarily completely reducible. The socle, Soc(V ), is the sum of all irreducible submodules of V . The head and socle of V are dual concepts; i.e. Soc(V ⇤)⇤ ⇠= V/Rad(V ).

A KS-module, V , is projective if any exact sequence

0 A B V 0 ! ! ! ! must split. Dually a KS-module V is injective if any exact sequence

0 V B A 0 ! ! ! ! must split.

Lemma 3.4. Let V be a ﬁnite-dimensional KS-module. Then

(i) There exists a ﬁnite-dimensional projective KS-module P such that V/Rad(V ) ⇠= P/Rad(P ).

(ii) This P is unique up to isomorphism.

(iii) V is a homomorphic image of P .

(iv) P/Rad(P ) ⇠= Soc(P ).

(v) dim(P ) is divisible by the order of a Sylowp subgroup of S.

(vi) If V = W U then there exists P and P such that P is projective and W is W U i a homomorphic image of PW and equivalently U of PU . CHAPTER 3. PRELIMINARY RESULTS 43

With the notation from Lemma 3.4, P is called the projective cover of V and the dual notion is called the injective hull. Importantly the dual concept of Lemma 3.4 (iii) is that any module can be embedded in an injective module. A consequence of Lemma 3.4 (iii) is that any finite-dimensional KS-module can be found as a quotient of some finite-dimensional projective module. It is also true that every finite-dimensional projective module is a direct sum of indecomposable projective modules, which are in 1-1 correspondence with the irreducible KS-modules as their projective covers. When searching for the restriction of L(G)toS we therefore only need to know the structure of the quotients of direct sums of projective indecomposables. Whilst that would be an infinite number of quotients to find, a useful lemma of Litterick [48] limits the possibilities to an often manageable finite set.

Lemma 3.5 (Litterick). Let S1,...,Sr be the irreducible KS-modules and P1,...,Pr the corresponding projective indecomposables. Let V be a self-dual KS-module having composition factors Si with multiplicity ri, such that V has no irreducible direct sum-

n(Pi) mands, and let P = P be the projective cover of V . Then n(P )+n(P ⇤) r i i i  i for all i. In particular,Ln(Pi) ri/2 when Si is self-dual.  With the restrictions the Lemma 3.5 places on the projective cover of L(G) we |S can now often calculate all such modules. We outline this method and its applications in more detail in Section 4.16.

We also introduce the following lemmas to complement Lemma 3.5 when determining ﬁxed vectors.

Lemma 3.6. Let V be a sum of projective indecomposables ⌃Pi whose trivial composition factors all of which are in Soc(V/Soc(V )). Then for any proper quotient Q of V if Q has no trivial submodules then there exists a projective quotient S of Q such that S contains all trivial composition factors of Q.

Proof. Let K be the kernel of the quotient map of Q, then Soc(K) Soc(V ). As  Soc(V )issemisimplethereexistsasubmoduleW Soc(V )suchthatSoc(V )=  Soc(K) W . As V is projective it must be injective and therefore is the injective hull of Soc(V ). By Lemma 3.4 there must exist P and P such that V = P P and P K W K W K CHAPTER 3. PRELIMINARY RESULTS 44

and PW are the injective hulls of Soc(K)andW respectively. As PK is the injective hull of Soc(K)weknowSoc(PK )=Soc(K)andthereforePK /K is a quotient of

PK /Soc(PK ). By our initial setup all of the trivial composition factors of PK /Soc(PK ) are in Soc(PK /Soc(PK )) and therefore if PK /K has no trivial submodules it must have no trivial composition factors. As PK /K has no trivial composition factors we have S = PW = Q/(PK /K)whereS is an injective (hence projective) quotient of Q containing all the trivial composition factors of Q, as required.

Lemma 3.7. Let H be a ﬁnite group and let V be a GF (p)H-module, p a prime. For v V , set K = Stab (v) and U = vH . Then U is isomorphic to a quotient of the 2 H h i GF (p) permutation module with H acting on the right cosets of K in H. Chapter 4

Proof of Theorems 1.2 and 1.3

From this point on G is always E8(2) and all modules are deﬁned over GF (2) unless otherwise stated.

4.1 L2(7)

From Chapter 6 p71 we have that there are seven feasible decompositions of L(G)for embeddings of subgroups H ⇠= L2(7) in E8(2). By Lemma 3.3(i), decompositions (ii), (iv), and (vi) have a trivial submodule and therefore by Lemma 3.2 any subgroups with these decompositions lie in a maximal subgroup from Section 3.1.1. For the remaining decompositions we need to know the structure of the projective covers of the non-trivial irreducible modules of L2(7). By direct calculation in Magma we ﬁnd the following;

P ( )= ( ) , P ( )= ( ) and P ( )= . 2 2| 3 1 | 2| 3| 2 3 3| 2 1 | 3| 2| 3 4 4

From this it is clear that all trivial composition factors must be in the second socle layer and so Lemma 3.6 applies. We can therefore conclude that if there are no trivial submodules then there exists a projective quotient containing all trivial composition factors, so there must be at least 5 composition factors of dimension 3 for each trivial. As in none of the remaining cases are there enough 3-dimensional composition factors, we may conclude all remaining feasible decompositions must have a trivial submodule and therefore that any subgroup H ⇠= L2(7) of E8(2) cannot be maximal as required.

45 CHAPTER 4. PROOF OF THEOREMS 1.2 AND 1.3 46

4.2 L2(11)

The situation for L2(11) is very similar to that of L2(7). From Chapter 6 p76 we have that there are six feasible decompositions of L(G)forembeddingsofsubgroups

H ⇠= L2(11) in E8(2). By Lemma 3.3(i) decompositions (v) and (vi) have a trivial submodule and therefore by Lemma 3.2 any subgroups with these decompositions lie in a maximal subgroup from Section 3.1.1. For the remaining decompositions we need to know the structure of the projective covers of the non-trivial irreducible modules of

L2(11). By direct calculation in Magma we ﬁnd the following;

P ( )= ( ) , P ( )= and P ( )= 2 2| 1 1 | 2 3 3| 3 4 4

From this it is clear that if there is no trivial submodule all trivial composition factors must be in the second socle layer and so Lemma 3.6 applies. This implies that if there is no trivial submodule then for each trivial composition factor there must be at least as many 2 and trivials, which allows us to know feasible decompositions (ii), (iii) and

(iv) must have a trivial submodule and therefore that any subgroup H ⇠= L2(11) of

E8(2) cannot be maximal as required. The remaining feasible decomposition, (i), is dealt with in [6] in which all subgroups of E8(2) with such a fusion are found, up to conjugacy, along with a suitable proper subgroup containing them.

4.3 L2(13)

From Chapter 6 p77 we have that there are six feasible decompositions of L(G)for embeddings of subgroups H ⇠= L2(13) in E8(2). By Lemma 3.3(i) and (iii) all decompositions have a trivial submodule and therefore by Lemma 3.2 any subgroups with these decompositions lie in a maximal subgroup from Section 3.1.1. L2(13) therefore cannot be a maximal subgroup of E8(2).

4.4 L2(17)

From Chapter 6 p78 we have that there are twenty feasible decompositions of L(G) for embeddings of subgroups H ⇠= L2(17) in E8(2). In Table 3.1.4 we may note that in E8(2) all elements of order 9 powered into elements in 3CE8(2) and therefore there CHAPTER 4. PROOF OF THEOREMS 1.2 AND 1.3 47 can be no embeddings corresponding to decompositions (i), (ii) and (ix) - (xvi). By Lemma 3.3(i) decompositions (iii) - (vi) and (xvii) - (xx) have a trivial submodule and therefore by Lemma 3.2 any subgroups with these decompositions lie in a maximal subgroup from Section 3.1.1. This leaves decompositions (vii) and (viii), for which we will need to know the structure of the projective covers of the non-trivial irreducible modules of L2(17). By direct calculation in Magma we ﬁnd the following;

P ( )= , 2 2| 1| 3| 1| 2| 1| 3| 1| 2| 1| 3| 1| 2| 1| 3| 1| 2

P ( )= , 3 3| 1| 2| 1| 3| 1| 2| 1| 3| 1| 2| 1| 3| 1| 2| 1| 3

P (4)=4 and P (5)=5

We know from Lemma 3.5 that the restriction, V, of L(G) to a subgroup H ⇠= L2(17) must be a quotient of some direct sum, D, of these projective covers. By considering the number of composition factors of 1 in P (2)andP (3) and in the decompositions

(vii) and (viii), we see that in these cases we see that there must be at least two P (2) or P (3)inthedirectsumD, implying that there are at least two composition factors of dimension 8 in the top of V . As, by assumption, there are no trivial quotients of V , then rad(V )containsasmanytrivialcompositionfactorsasV with at most ﬁfteen composition factors of dimension 8, which by Lemma 3.3(i) gives us a trivial submodule. As all feasible decompositions have now been shown to imply a ﬁxed vector we can say that L2(17) cannot be a maximal subgroup of E8(2).

4.5 L2(25)

From Chapter 6 p84 we have that there are eight feasible decompositions of L(G)for embeddings of subgroups H ⇠= L2(25) in E8(2). By Lemma 3.3(i), (iii) decompositions (ii) - (viii) have a trivial submodule and therefore by Lemma 3.2 any subgroups with these decompositions lie in a maximal subgroup from Section 3.1.1. For the remaining case there are no trivial composition factors and it therefore cannot fix a vector. In [6] it is found that there are no embeddings H with fusion corresponding to case (i) in E (2) on the assumption that for V = L(G) , R Syl (H)ande H of order 4 8 |H 2 5 2 the following holds; dimCV (R)=8anddimCV (e)=16.Thefirstoftheseproperties can be checked on each composition factor linearly as R is coprime to 2. In order | | CHAPTER 4. PROOF OF THEOREMS 1.2 AND 1.3 48 to check that the second property must hold we need to know the structure of the projective covers of 4 and 5. By direct calculation in Magma we find the following;

P ( )= and P ( )= 4 4| 4| 4| 4 5 5

From Lemma 3.5 we know that any direct summands of V not isomorphic to an irreducible L2(25)-module must be quotients of a direct sum of no more than two

P (4). We find by direct calculation all such quotients and find the corresponding fixed spaces of e and R. The results split into three cases; V 144 26 26 26 26 ⇠ | | and V 144 26 26 26 26 and V 144 26 26 26 26, with dimC (e) ⇠ | ⇠ V 64, 66 and 68 respectively. From the atlas [16] and Chapter 6 we see that L2(25) has elements of order 12 whose 3rd power is conjugate to e and whose 4th power is in 3D , therefore e C (x) = 3 U (2) for some x 3D . Using the E8(2) 2 E8(2) ⇠ ⇥ 9 2 E8(2) Magma command LMGRadicalQuotient we get H ⇠= U9(2) as a permutation group as well as a homomorphism, from C (x) H. In H we find conjugacy class E8(2) ! representatives for all classes of order 4 and 12 and find their inverse image under , giving us representatives of all classes of order 12 in H. Only a single class of these elements has the correct fixed space, 128, when taken to the 6th power and therefore it is in the 3rd power of this class that e must lie. By direct calculation we are therefore able to determine that dimCV (e)=16.

4.6 L2(27)

From Chapter 6 p86 we have that there are two feasible decompositions of L(G)for embeddings of subgroups H ⇠= L2(27) in E8(2). By Lemma 3.3(i) all decompositions have a trivial submodule and therefore by Lemma 3.2 any subgroups with these decompositions lie in a maximal subgroup from Section 3.1.1. L2(27) therefore cannot be a maximal subgroup of E8(2). CHAPTER 4. PROOF OF THEOREMS 1.2 AND 1.3 49

4.7 L4(3)

From Chapter 6 p95 we have that there are four feasible decompositions of L(G)for embeddings of subgroups H ⇠= L4(3) in E8(2). By Lemma 3.3(i) and (iii) all decompositions have a trivial submodule and therefore by Lemma 3.2 any subgroups with these decompositions lie in a maximal subgroup from Section 3.1.1. L4(3) therefore cannot be a maximal subgroup of E8(2).

4.8 U3(3)

From Chapter 6 p83 we have that there are four feasible decompositions of L(G)for embeddings of subgroups H ⇠= U3(3) in E8(2). By Lemma 3.3 (i) and (iii) all decompositions have a trivial submodule and therefore by Lemma 3.2 any subgroups with these decompositions lie in a maximal subgroup from Section 3.1.1. U3(3) therefore cannot be a maximal subgroup of E8(2).

4.9 G2(3)

From Chapter 6 p94 we have that there are two feasible decompositions of L(G)for embeddings of subgroups H ⇠= G2(3) in E8(2). By Lemma 3.3 (i) and (iii) both decompositions have a trivial submodule and therefore by Lemma 3.2 any subgroups with these decompositions lie in a maximal subgroup from Section 3.1.1. G2(3) therefore cannot be a maximal subgroup of E8(2).

4.10 M11

From Chapter 6 p85 we have that there are three feasible decompositions of L(G) for embeddings of subgroups H ⇠= M11 in E8(2). By Lemma 3.3 (i) and (iii) all decompositions have a trivial submodule and therefore by Lemma 3.2 any subgroups with these decompositions lie in a maximal subgroup from Section 3.1.1. M11 therefore cannot be a maximal subgroup of E8(2). CHAPTER 4. PROOF OF THEOREMS 1.2 AND 1.3 50

4.11 M12

From Chapter 6 p90 we have that there are three feasible decompositions of L(G) for embeddings of subgroups H ⇠= M12 in E8(2). By Lemma 3.3 (i) and (iii) all decompositions have a trivial submodule and therefore by Lemma 3.2 any subgroups with these decompositions lie in a maximal subgroup from Section 3.1.1. M12 therefore cannot be a maximal subgroup of E8(2).

4.12 L3(8)

From Chapter 6 p106 we have that there is one feasible decompositions of L(G)for embeddings of subgroups H ⇠= L3(8) in E8(2). By Lemma 3.3 (i) both decompositions have a trivial submodule and therefore by Lemma 3.2 any subgroups with these decompositions lie in a maximal subgroup from Section 3.1.1. L3(8) therefore cannot be amaximalsubgroupofE8(2).

4.13 F4(2)

From Chapter 6 p114 we have that there are four feasible decompositions of L(G)for embeddings of subgroups H ⇠= F4(2) in E8(2). By Lemma 3.3 (i) all decompositions have a trivial submodule and therefore by Lemma 3.2 any subgroups with these decompositions lie in a maximal subgroup from Section 3.1.1. F4(2) therefore cannot be amaximalsubgroupofE8(2).

2 4.14 F4(2)0

From Chapter 6 p96 we have that there are two feasible decompositions of L(G)for

2 embeddings of subgroups H ⇠= F4(2)0 in E8(2). By Lemma 3.3 (i) all decompositions have a trivial submodule and therefore by Lemma 3.2 any subgroups with these de-

2 compositions lie in a maximal subgroup from Section 3.1.1. F4(2)0 therefore cannot be a maximal subgroup of E8(2). CHAPTER 4. PROOF OF THEOREMS 1.2 AND 1.3 51

3 4.15 D4(2)

From Chapter 6 p98 we have that there are four feasible decompositions of L(G)

3 for embeddings of subgroups H ⇠= D4(2) in E8(2). By Lemma 3.3 (i) and (iii) all decompositions have a trivial submodule and therefore by Lemma 3.2 any subgroups

3 with these decompositions lie in a maximal subgroup from Section 3.1.1. D4(2) therefore cannot be a maximal subgroup of E8(2).

4.16 L3(3)

From Chapter 6 p83 we have that there are ﬁve feasible decompositions of L(G)for embeddings of subgroups H ⇠= L3(3) in E8(2). By Lemma 3.3(i) decompositions (iii) - (v) have a trivial submodule and therefore by Lemma 3.2 any subgroups with these decompositions lie in a maximal subgroup from Section 3.1.1. The remaining two cases will now be approached separately.

4.16.1 Case (ii)

For this case we assume there exists a subgroup H ⇠= L3(3) in E8(2) following fusion pattern ii. From the atlas [16] we can see that H has maximal subgroup of the form

13 : 3 which is the normailser of a Sylow13 subgroup. From Chapter 6 we have that all elements of order 13 in H are in the 13BE8(2) class. Therefore as we are only looking for such groups up to conjugacy without loss of generality we may choose an element x in 13B . We therefore choose one in N 132.(12 GL (3)), the normaliser of a E8(2) ⇠ ⇤ 2 Sylow13 subgroup of G constructed in [6]. We then use the FindCentProc outlined in Section 3.2 to get C (x), and, combined with an element y N of order 3 which E8(2) 2 acts non-trivially on x, we get a group L containing all possible 13 : 3 containing x. By checking the ﬁxed space dimensions of conjugacy class representatives of L we

ﬁnd there are no elements of 3CE8(2) in L, which is a contradiction with our initial assumption as all elements of order 3 in fusion case (ii) are in 3CE8(2). CHAPTER 4. PROOF OF THEOREMS 1.2 AND 1.3 52

4.16.2 Case (i)

For case (i) we will need to know the structure of the projective covers of the non-trivial irreducible modules of L3(3). By direct calculation in Magma we ﬁnd the following;

P ( )= ( ) , 2 2| 1 2 | 3| 1| 2

P ( )= ( ) ( ) ( ) and 3 3| 1 3 | 2 3 | 1 3 | 3

P (4)=4

We now assume that there exists a subgroup H ⇠= L3(3) in E8(2) following fusion pattern (i). Applying Lemma 3.5 we know that for any direct summand D of V = L(G) , V/Rad(V )isthedirectsumofsomesubsetofthemultiset , , , , . |H { 2 3 3 3 3} Again we split this analysis into two: we ﬁrst tackle those subsets S with four or more elements, , , , , , , , and , , , , . In each case we may { 3 3 3 3} { 2 3 3 3} { 2 3 3 3 3} assume that D is a submodule of PD = ⌃P (i)fori in S whose socle coincides with that of PD. We shall now show that no such D can exist.

Proposition 4.1. Let H ⇠= L3(3) be a subgroup of E8(2) following fusion pattern (i) which does not ﬁx a vector on L(G). Then let D be an indecomposable direct summand of V = L(G) for which D/Rad(D)andSoc(D)areoneof |H

(a) , 2 3 3 3

(b) or 3 3 3 3 (c) . 2 3 3 3 3 Then there are exactly 4 trivial composition factors of D.

Proof. We shall assume the contrary that there exists another indecomposable summand A with a trivial composition factor. We note that as D is indecomposable Rad(D) Soc(D), therefore dim(D) > 124, which implies that D is self-dual because ◆ V is. In each case we now know an upper bound for the composition factors of each isomorphism type for A. For case (a) A has at most one 2 and two 3, which by

Lemma 3.5 means A is a quotient of P (3). By direct calculation in Magma we can conﬁrm that there are no quotients of P (3)withnotrivialsubmodulesandatthe CHAPTER 4. PROOF OF THEOREMS 1.2 AND 1.3 53

required composition factors. Similarly for case (b) A has at most three 2 and no

3, which by Lemma 3.5 means A is a quotient of P (2). Again by direct calculation in Magma we can conﬁrm that there are no quotients of P (2)withnotrivial submodules and at the required composition factors. This leaves case (c) for which A can have at most one 2 as non-trivial composition factors, which by Lemma 3.5 implies A is irreducible, a contradiction. Having exhausted the possibilities for A we can now conclude that such a direct summand cannot exist, and therefore the proposition holds.

In what follows we shall ﬁnd all indecomposable modules m whose composition factors are consistent with fusion case (i) with Soc(m)adirectsumofthreeorfewer irreducibles, and so dually pick up all those with m/Rad(m)likewise.Wetherefore need to determine the possibilities for indecomposable modules m whose composition factors are consistent with fusion case (i) with four or more irreducible modules in both Soc(m)andm/Rad(m), i.e. those covered by Proposition 4.1. We again note that such m must be self-dual as above. In each of the three cases we apply the code in Appendix A whose function shall now be brieﬂy explained

The main building block of the method for finding the indecomposable modules comprises taking minimal submodules of ever smaller quotients and collecting their inverse images under the quotient map. Let S be a submodule of some direct sum of projective indecomposables DM, whose composition factors are a subset of those in agivenfeasibledecomposition.Thenwefindallminimalsubmodulesofthequotient DM/S and check first whether their inverse images also have compatible composition factors, and secondly whether they have a trivial quotient. We then keep a set of all submodules, up to isomorphism, which pass the first check to repeat the process and any that pass both sets are the indecomposable modules we are trying to find. By repeating this process starting with the zero submodule and continuing until no submodules pass the first check we generate a list of all possible indecomposable submodules of DM that could be a direct summand of our feasible decomposition such that no non zero vector is fixed.

As an extension of this method we can be more speciﬁc and look only for those CHAPTER 4. PROOF OF THEOREMS 1.2 AND 1.3 54 modules with a given socle and head. To do so we simply start the previous method at the required socle rather than at the zero module and then collect all modules that pass the ﬁrst check with the composition factors required for the top removed from the feasible decomposition. Finally, for each case remaining we look for all submodules isomorphic to the required top in the quotient.

In case (a) we get 26 intermediate submodules and 70 final submodules, in case (b) we get 4 intermediate submodules and 16 final submodules and in case (c) we get 3 intermediate submodules and 3 final submodules. Checking all the final submodules we find that none of them are self-dual. We can therefore conclude that there are no direct summands m of V = L(G) with four or more composition factors in both |H Soc(m)andm/Rad(m).

We now turn to the case where either Soc(m)orm/Rad(m)hasthreeorfewer composition factors, and, by duality, it is sucient to look just at Soc(m). We ﬁrst look for all submodules up to isomorphism of P ( ) P ( ) P ( ). We do this using 2 3 3 the code in Appendix A, then ﬁltering out reducible submodules. The results are shown in Table 4.16.1 with the following labelling; the module name is the dimension of the module with an arbitrary labelling to distinguish those of equal dimension, we use SD to denote a self-dual module and N/A to denote a non-self dual module which when added to its dual has too many of a particular composition factor to be in fusion case (i) and which can therefore be ignored when analysing the results below. We also repeat this process for P ( ) P ( ) P ( ); however, we get no new 3 3 3 modules with 26, 39a, 52, 65a, 78a, 91a, 105, 117a and 144 reoccurring in this case.

From the feasible decomposition associated with fusion case (i) we know that there are more trivial composition factors than there are of composition factors isomorphic to 2, therefore there is at least one direct summand where the same is true. Looking through Table 4.16.1 we see that there are only two such modules, 105 and 144. For 105 we now need to ﬁnd all combinations of modules from Table 4.16.1 whose direct sum has composition factors exactly 3 , 3 , 4 . We ﬁrst note that this direct sum { 1 2 3} cannot contain a 2 module as then we would once again have more trivial composition CHAPTER 4. PROOF OF THEOREMS 1.2 AND 1.3 55

Module Dual #1 #2 #3 Module Dual #1 #2 #3 12 SD 0 1 0 129a N/A 1 2 4 24 SD 0 2 0 129b N/A 1 2 4 26 SD 0 0 1 130a SD 2 2 4 39a 39b 1 1 1 130b N/A 2 2 4 39b 39a 1 1 1 130c SD 2 2 4 51a 51b 1 2 1 130d N/A 2 2 4 51b 51a 1 2 1 130e SD 2 2 4 52 SD 0 0 2 142 N/A 2 3 4 64 SD 1 3 1 143a N/A 3 3 4 65a 65b 1 1 2 143b N/A 3 3 4 65b 65a 1 1 2 144 SD 2 1 5 77a 77b 1 2 2 156a N/A 2 2 5 77b 77a 1 2 2 156b N/A 2 2 5 78a SD 0 0 3 156c N/A 2 2 5 78b SD 2 2 2 156d N/A 2 2 5 78c SD 2 2 2 169a N/A 3 3 5 78d SD 2 2 2 169b N/A 3 3 5 90 N/A 2 3 2 169c N/A 3 3 5 91a 91b 1 1 3 182a N/A 2 2 6 91b 91a 1 1 3 182b SD 2 2 6 103a N/A 1 2 3 182c N/A 2 2 6 103b N/A 1 2 3 195a N/A 3 3 6 104a SD 2 2 3 195b N/A 3 3 6 104b N/A 2 2 3 195c N/A 3 3 6 104c N/A 2 2 3 208a N/A 2 2 7 104d SD 2 2 3 208b N/A 2 2 7 105 SD 1 0 4 221a N/A 3 3 7 116 SD 2 3 3 221b N/A 3 3 7 117a 117b 1 1 4 234 SD 2 2 8 117b 117a 1 1 4 247 N/A 3 3 8 117b N/A 3 3 3

Table 4.16.1: Indecomposable L3(3)-modules

factors than 2 factors and there can be no more than one of 105 or 144. There are no self-dual modules or pairs of dual modules with an odd number of composition factors isomorphic to 2 and less than five composition factors isomorphic to 3 so it is impossible to find a combination of the form required for 105. For 144 we need to find all combinations of modules from Table 4.16.1 whose direct sum has composition factors exactly 2 , 2 , 3 . There are no self-dual modules or pairs of dual modules { 1 2 3} with trivial composition factors and less than two composition factors isomorphic to , so the combinations we are looking for are all of the form A i for some i 0 2 3 and A from Table 4.16.1. By checking which modules can be A we find the following possibilities for V . 144 39 39 ,144 78 , a b 3 b 3 CHAPTER 4. PROOF OF THEOREMS 1.2 AND 1.3 56

144 78 ,144 78 , c 3 d 3 144 104 and 144 104 a d We now check the dimension of the fixed point space of an involution on these six modules and compare it to that of E8(2) from Section 3.2. In the first five cases we get 126 and in the last case we get 128. There is no conjugacy class of involutions in E8(2) with fixed point space dimension 126 but there is one with dimension 128, 2D . This implies that if there exists a subgroup H L (3) in E (2) following E8(2) ⇠= 3 8 fusion pattern (i) then V = L(G) = 144 104 . |H ⇠ d

4.17 L3(4)

From Chapter 6 p87 we have that there are two feasible decompositions of L(G)for embeddings of subgroups H ⇠= L3(4) in E8(2). In [6] it is found that no there are no 2 elementary abelian subgroups of order 3 in E8(2) with compatible ﬁxed space to that of feasible decomposition (i). We therefore proceed with feasible decomposition (ii) in a similar fashion to that applied to L3(3). We assume that there is a subgroup H ⇠= L3(4) in E (2) compatible with feasible decomposition (ii), such that M = L(G) has no 8 |H trivial submodules or quotients. From Lemma 3.5 we know that M is a submodule of P ( ) P ( ). We ﬁnd the following irreducible modules: 2 3 From the feasible decomposition we are working with we know there is a submodule of dimension 56 containing all composition factors not isomorphic to 5. We can therefore ignore all indecomposable modules of dimension > 28 which are not self- dual. We also note that there are no modules with more trivial composition factors than composition factors of dimension 9, which given there are equal numbers in our feasible decomposition implies they are equal in all indecomposable summands. This allows us to ignore modules 34 and 54b, leaving the irreducibles and 20, 36b and 56b. This now leaves us with three types of possibilities for V = L(G) . |H

20 20 4 5 5 5

20 36 b 5 5 5 56 b 5 5 5 CHAPTER 4. PROOF OF THEOREMS 1.2 AND 1.3 57

Module Dual Multiplicity #1 #2 #3 #4 #5 9a 9b 1 0 1 0 0 0 9b 9a 1 0 0 1 0 0 16 SD 1 0 0 0 1 0 20 SD 2 2 1 1 0 0 34 SD 6 0 1 1 1 0 36a 36c 3 2 2 0 1 0 36b SD 6 2 1 1 1 0 36c 36a 3 2 0 2 1 0 45a 45b 42 2 2 1 1 0 45b 45a 42 2 1 2 1 0 54a 54c 18 2 2 2 1 0 54b SD 9 2 2 2 1 0 54c 54a 18 2 2 2 1 0 56a 56c 57 4 2 2 1 0 56b SD 27 4 2 2 1 0 56c 56a 57 4 2 2 1 0 64 SD 1 0 0 0 0 1

Table 4.17.1: Indecomposable L3(4)-modules

Unfortunately there is little further that can be determined. When checking the fixed space of involutions the modules of type 56b split into two cases, one with fixed space dimension 30 and one with fixed space dimension 32. Of the possibilities given above, the only cases with fixed space dimensions incompatible with those of E8(2) are the 56 where the 56 is one with fixed space dimension of an involution 30. b 5 5 5 b

4.18 U3(8)

From Chapter 6 p102 we have that there are two feasible decompositions of L(G)for embeddings of subgroups H ⇠= U3(8) in E8(2).

4.18.1 Case (ii)

For this case we assume there exists a subgroup H such that the restriction M = L(G) has no trivial quotients or submodules and is compatible with feasible decom- |H position (ii). There exists a minimal submodule S subject to the condition that all trivial composition factors are in S and S also has no trivial quotients or submodules. As M is self dual and the only irreducible module of U (8) in with dim H1(H, ) =0 3 i 6 is 3 then there must be a quotient of S isomorphic to 3. Taking the kernel, K,ofthis CHAPTER 4. PROOF OF THEOREMS 1.2 AND 1.3 58 quotient we now have a submodule of S with 8 trivial composition factors and only one 3 composition factor therefore by Lemma 3.3 there exists a trivial submodule of dimension at least 2. This is a contradiction and therefore no subgroup compatible with feasible decomposition (ii) can be maximal in E8(2) by Lemma 3.2.

4.18.2 Case (i)

From the atlas [16] we know that U3(8) has a maximal subgroup, M, isomorphic to 3 L (8), and by direct calculation combined with the fusion information in Chapter 6 ⇥ 2 we know that any element of order 3 in the center of M is in the 3D conjugacy class of

E8(2). Without loss of generality we may choose an element, x,in3DE8(2) and ﬁnd its centraliser, C (x) = 3 U (2). We now wish to ﬁnd up to conjugacy all subgroups E8(2) ⇠ ⇥ 9 of this U9(2) isomorphic to L2(8) which we shall do using the methods outlined in

Chapter 5. We first find an isomorphism from our U9(2) to the natural 43605-degree permutation representation of it using LMGRadicalQuotient. We must then find all cuspidal classes of subgroups of order 7 in U9(2), which is simple to do as the Sylow7 subgroup of U9(2) is of order 7, and so we get two parabolics with a single cuspi- 30 dal class each, P (1, 2, 7, 8) and P (2, 3, 6, 7), both with shape [2 ]:3.L3(4).3. Here, as in Chapter 5, we use P (xi,...,xj)todenotetheparabolicwithsimpleroots(xi,...xj).

In order to ﬁnd these parabolics within our permutation representation of U9(2) we use the following stabilisers to form the maximal parabolics and then take appropriate intersections. We use PR to denote the permutation group isomorphic to U9(2) and ⌦ as the PR-set it acts upon.

P 1=P (2, 3, 4, 5, 6, 7) [215]:U (2) = Stab (1). ⇠ 7 PR Let T Sylow (P 1) and let SST be the fourth term in the upper central series of T , 2 2 4 of order 26. We then get

P 2=P (1, 2, 4, 5, 7, 8) [224]:U (2) 3.L (4).3=Stab (Fix (SST )). ⇠ 3 ⇥ 3 PR ⌦ 4

Letting T 0 be the derived subgroup of T and T 00 the derived subgroup of T 0 we get the ﬁnal two maximal parabolics as

24 P 3=P (1, 3, 4, 5, 6, 8) [2 ] : Alt U (2) = Stab (Fix(T 0)) and ⇠ 5 ⇥ 5 PR CHAPTER 4. PROOF OF THEOREMS 1.2 AND 1.3 59

24 P 4=P (1, 2, 3, 5, 6, 7, 8) [2 ]:L (4) = Stab (Fix(T 00)). ⇠ 4 PR

Having applied the method to these subgroups and found two sets of copies of L2(8) we can check conjugacy in U9(2) and get ﬁve representative subgroups, 1 5.

1 3

For this case we assume there exists a subgroup i

For this case we assume there exists a subgroup 4

We now note that S ,aSylow subgroup of the chosen maximal 3 L (8) in U (2), is 2 2 ⇥ 2 9 elementary abelian of order 8 and there exists an element y of order 4 in CU9(2)(S2)such that y2 S . We therefore now search for all such elements in C (Sylow ( )). For 2 2 E8(2) 2 5 5 we let g1,g2,g3 be the three generators of S2 and use CentraliserOfInvolution to ﬁnd C (g ), which as g 2D we know must be of shape [284]:Sp (2), E8(2) 1 1 2 E8(2) 8 therefore we can check we have the whole centraliser as CentraliserOfInvolution is Monte Carlo. We then use LMGRadicalQuotient, with quotient map q, to ﬁnd just

2 the Sp8(2) and map S2 into this quotient. When we do this we find q(S2)isoforder2 and can calculate C (q(S )) and find it to be of order 210 3, and that the Sylow Sp8(2) 2 · 2 of the centraliser is self-normalising. Taking the inverse image of this centraliser we get a group C1oforder294 3whichcontainsC (S ). As we are only interested is · E8(2) 2 elements of order 4 we can take just the Sylow2sofC1, of which there are 3 by Sylow’s theorems. We know any element of C1thatcentralisesS2 must stabilise the fixed space, E,ofS2, so we use the command UnipotentStabiliser to find the stabiliser of E in each of the Sylow subgroups. We can now directly calculate the centraliser in these stabilisers giving us a collection of all unipotent elements which centralise S2. We now proceed to create an isomorphism to a PC group and make a transversal of the center in these centralisers and check each element of order 4 as to whether it squares into S2. When we do this none of the elements have the required properties so there are no copies of U3(8) generated in this case. CHAPTER 4. PROOF OF THEOREMS 1.2 AND 1.3 61

4.19 U3(16)

From Chapter 6 p122 we have that there are two feasible decompositions of L(G)for embeddings of subgroups H ⇠= U3(16) in E8(2). We assume that there exists a subgroup H = U (16) in G. We may infer that L(G) is semisimple as it is self dual ⇠ 3 |H with two non-isomorphic self dual composition factors, therefore there is exactly one possible restriction in each case that we shall denote M1 and M2. From the atlas [16], U (16) has a maximal subgroup S = 17 L (16) where any element in the center 3 ⇠ ⇥ 2 of S is in a U3(16) conjugacy class that fuses to 17CDE8(2) for both case (i) and (ii). As for x 17CD we know C (x) = 17 L (16) so C (x)=C (x)ifx H.We 2 E8(2) G ⇠ ⇥ 2 G H 2 may now choose up to conjugacy an x 17CD and must ﬁnd its centraliser in G. 2 E8(2)

92 We choose an x such that it lies in a Levi complement, L,ofthe[2 ]:L8(2) standard parabolic. Using the method from Section 5.1 we can search for the subgroup

S4 in the 2-core such that S4 is a four-dimensional trivial module for x. Having found this, we ﬁnd the centraliser CL(x)oforder3.5.17 and choose an element, y, of order

15 in CL(x)thatactsirreduciblyonS4 (this should be all of them). Now in order to generate L2(16) from S4 and y we require an inverting involution for y. Using the method from Section 3.2 we can find an inverting involution and extended centraliser, EC, for y5 3D and in the radical quotient EC of this group we search for an 2 E8(2) inverting involution in the normaliser in EC of the image of y3 5B . We can also 2 E8(2) 5 find C (y) C (x )inasimilarfashionandsohavetheextendedcentraliserC⇤ (y). G  G G We now simply find any inverting involution in CG⇤ (y)thatcommuteswithx and we have full generation for CG(x).

We now compare the restriction of our two possible U3(16) modules M1andM2 to the maximal 17 L (16) with the restricition L(G) that we just obtained. For ⇥ 2 |CG(x) both M and M the composition factors of M have dimension 1 2 i|S 2,2,2,2,24,24,48,48,48,48; whereas for L(G) the composition factors have dimen- |CG(x) sion 1,1,1,1,1,1,1,1,8,8,16,16,16,16,32,32,32,64. This is a contradiction so we may conclude that there are no subgroups isomorphic to U3(16) in E8(2). This, along with Tables 3.1.3 and 3.1.5 and Lemma 3.1, completes the proof of Theorem 1.1. Chapter 5

Proof of Theorem 1.4

5.1 Methodology

For the many groups in Table 3.1.2 the methods used for the groups in the previous section are either insucient to determine maximality, or too computationally dicult to complete, so we will now outline an alternative approach. In this method we ﬁnd all embeddings of the groups in question up to conjugacy in E8(2) and then check individually if they are maximal.

Lemma 5.1. Let H be a group isomorphic to L (2n) n 3, 4, 5, 6, 7 and H 2 2 { } S Syl (H). Then S is elementary abelian of order 2n, there exists an element 2 2 x N (S) of order 2n 1 and an involution t that inverts x such that N (S)= S, x 2 H H h i and H = S, x, t . Furthermore x acts irreducibly on S. h i Proof. This can be easily checked by direct calculation in Magma.

Theorem 5.1 (Borel-Tits Theorem). Let H be a simple linear algebraic group defined over an algebraically closed field of characteristic p =0. Let be a Frobenius morphism 6 on H and H the fixed point group of . Then let U be a non-identity p-subgroup of H . There then exists a parabolic subgroup P such that N (U) P and U O (P ). H ✓ ✓ p Proof. See Burgoyne, Williamson [12].

We now deﬁne a cuspidal class in an analogous fashion to those of Coxeter groups. A cuspidal class of G, a group of Lie type, is a conjugacy class C of G such that

62 CHAPTER 5. PROOF OF THEOREM 1.4 63

C H = for all parabolic subgroups H of G. \ ;

Lemma 5.2. Let G be a ﬁnite algebraic group with a parabolic subgroup H, p a prime, S (G) a Sylow subgroup of G and S (H) a Sylow subgroup of H. Then if S (G) = p p p p | p | S (H) there are no cuspidal classes of G of order p. | p |

Proof. By Sylow’s theorems we know that Sp(H)isaSylowp subgroup of G.For any element x of order p we can therefore ﬁnd a Sylowp subgroup of G containing it and an element conjugating that subgroup to a Sylowp subgroup of H. This element conjugates x into H thereby proving the lemma.

By Lemma 5.1 for a subgroup H G, H = L (2n)wehaveS Syl (H)isele-  ⇠ 2 2 2 mentary abelian order 2n. As N (S) N (S)wecanuseTheorem5.1tosaythat H  G N (S) P for some parabolic subgroup P G, which as we are interested up to H   conjugacy can be chosen to be a standard parabolic. If the element x of order 2n 1 is in a cuspidal class then P is as small as possible; however, if it is not, then within P we can conjugate H into a smaller parabolic P 0 P whilst maintaining S O (P 0).   2 By choosing all parabolic subgroups with cuspidal classes we get, for a given n>3, a set ⇧ of parabolic subgroups for which all subgroups N (S) G for H = L (2n)exist H  ⇠ 2 as a subgroup of at least one parabolic. We then proceed to search each parabolic for all such NH (S).

Within a chosen parabolic subgroup P ⇧ we can choose representatives of the 2 cuspidal classes of its Levi complement and study the action of these elements on

O2(P ). Let x be such an element, then by Lemma 5.1 x acts irreducibly on S, so n we are looking for all subgroups of order 2 in O2(P )thatarefixedundertheac- tion of x. To do this we first find the action of x on the quotient O2(P )/F (O2(P )) where F (O2(P )) is the Frattini subgroup of Op(P )usingtheMagma code GModule, and find the preimage of all irreducible n-dimensional modules. As any possible S either projects fully or trivially into this quotient, S is then a subgroup of at least one of these preimages, so this process is repeated until the groups are small enough to allow all subgroups of order 2n fixed by x to be found. Due to the number of submodules of a direct sum of isomorphic irreducible modules, when the action of x CHAPTER 5. PROOF OF THEOREM 1.4 64

on O2(P )/F (O2(P )) has isomorphic irreducibles in its direct sum decomposition we instead take the preimage of the direct sum of these irreducibles, which will contain the preimages of all of the submodules we would otherwise have to ﬁnd. Once all the irreducible subgroups of order 2n have been found we then must run through all

n inverting involutions in CG⇤ (x)tocheckiftheygenerateL2(2 ).

There are several stumbling blocks to the smooth running of the method outlined above that we shall now explain alongside the Magma code, found in Appendix A, that was used both for the method as a whole as well as the problem cases. In the first section of code we approach the general situation where we have some 2-group P and we know nothing of the structure of P or the action of x on P . Here we find the Frattini subgroup F (P )andnotethatthequotientP/F(P )iselementaryabelian and normal under the action of x, therefore can be considered as an -module using the GModule command. We are interested in n-dimensional minimal submodules of this module as any irreducible subgroup of order 2n will either be in the kernel of the quotient or project fully as such an irreducible module. We find a direct sum decomposition of the module and split the n-dimensional components into isomorphism classes; as a submodule s of a direct sum of irreducibles is a submodule of the direct sum of only those components isomorphic to s. Having found these direct sums we return to the group setting, where we now have a collection of smaller groups in which all irreducible subgroups reside. At this point we must split further calculation into two cases; one where there was more than one isomorphism class, and the other where there was only one. In the first case the subgroups we found are proper and we can repeat the process until we are left with just a collection of irreducible elementary abelian subgroups which we can then easily split up into groups of order 2n defined by the orbits of x on them. In the second case we must use a more computationally intense method.

In this second case repeating the method above would result in an inﬁnite loop where the original subgroup P would be repeatedly returned, so we need a method of splitting this into smaller subcases. First we choose an element from every minimal n-dimensional submodule of M = P/F(P ). We do this by choosing an ordering on CHAPTER 5. PROOF OF THEOREM 1.4 65

the direct sum decomposition of S, and fixing a non-zero vector wi in each summand. Then we work through the summands collecting all vectors which are of the form v = w v where v is any vector contained in the direct sum of all sumands with i i i j>i. By pulling these back into the group setting this allows us to find all irreducible subgroups, Hi, as required, however there will commonly be far more of these than is practical to then perform any of the calculations previously described. We therefore look for properties of the generating element, hi of each group to allow us to rule many of them out. For this we require a subgroup A F (P ) such that F (P )/A Z(P/A)   and F (P )/A is elementary abelian. This subgroup can be found by first taking the commutator subgroup C =[P, F(P )] which gives F (P )/C Z(P/A), and then find-  ing the Frattini subgroup of F (P )/C gives us the subgroup containing C such that

F (P )/A is elementary abelian. Taking some Hi found before we know A E F (P ) E Hi and for an irreducible subgroup s to project non-trivially into Hi/F (P )itmustalso do so into H/A and therefore we are only interested in Hi such that hi is an involution in Hi/A. This gives us an easy to check condition that frequently reduces the subcases drastically down to manageable proportions. In the cases that remain this extra condition implies that Hi/A is elementary abelian so we can proceed using our original method on each Hi.

Finally having found all the irreducible subgroups we need an ecient way of determining which pairs of an irreducible subgroup, s, and inverting involution generate

n acopyofL2(2 ). To do this we ﬁrst check that the elements of s are in the same

E8(2) conjugacy class by checking their ﬁxed space dimension. We then check the orders of products x t where x s, sieving out all those that give element orders · 2 n not in L2(2 ). Finally we check the number of composition factors of the restriction L(G) is consistent with a feasible decomposition. Having done this we generate the |s group s as a subgroup of GL248(2) and proceed from there.

In the following sections we shall use P (ij . . . )todenotetheparabolicsubgroup of G corresponding to the choice of roots xi,xj,.... CHAPTER 5. PROOF OF THEOREM 1.4 66

5.2 L2(128)

The Sylow-127 subgroup of G is cyclic of order 127, so without loss of generality we may choose a single 127-element in each parabolic that we have to search, for which all other 127-elements can be conjugated within the parabolic into the same Sylow- 127. The three standard parabolic subgroups containing elements cuspidal classes of order 127 are all isomorphic to L (2) and correspond to the roots 1, 3, 4, 5, 6, 7 , 7 { } 2, 4, 5, 6, 7, 8 and 3, 4, 5, 6, 7, 8 respectively. For each of these groups we need to { } { } choose an element of order 127, noted as x1,x2,x3 respectively, as well as the extended centraliser CG⇤ (xi)foreach.

From Table 3.1.4 the centraliser of an element of order 127 is Sym(3) 127. In ⇥ the case of P (134567) this centraliser can be found easily from the root elements as the longest root in the extended Dynkin diagram of E8 commutes with the whole parabolic and so generates the necessary Sym(3). Likewise in the P (245678) case, the group generated by the root 1 generates the necessary Sym(3). In the P (345678) case there is no such easy root construction; however, both it and P (245678) are contained in the maximal rank subgroup generated from the roots of the extended Dynkin dia- + gram without root 1, of shape ⌦16(2). Inside this group there exists both an element conjugating Sylow-127 subgroup of P (245678) containing x2 to a Sylow-127 subgroup of P (345678) and an involution which inverts x2. This allows us to generate CG⇤ (x2) and conjugate it to ﬁnd x3 and CG⇤ (x3). The inverting involution for x1 was found using the method from section X applied within a Levi complement of P (1234567) ⇠= E7(2), giving us CG⇤ (x1). These CG⇤ (xi) contain four inverting involutions, so for each irreducible 27 subgroup found, we need to check with these four involutions to check for generation of an L2(128).

Using the method outlined above and the code in Appendix A we proceed to check each of the parabolic subgroups in turn. In each case there are no large bad cases as deﬁned above, so the analysis runs smoothly and quickly and returns four subgroups in the case of P (134567) and two each for P (245678) and P (345678). We can directly calculate the dimension of the ﬁxed space of these groups on V = L(G) , and we get |H CHAPTER 5. PROOF OF THEOREM 1.4 67

1fortwoofthegroupsinthecaseofP (134567) and 6 for all other groups. We can therefore use Lemma 3.2 to show that these groups are not maximal as required.

5.3 L2(32)

From Chapter 6 p105 we know there is only one feasible decomposition of L2(32) on L(G), from which we know that all elements of order 31 in a subgroup isomorphic to L2(32) must lie in the 31ABC class(es) in E8(2). There are 6 standard parabolics containing cuspidal classes of elements of order 31, all isomorphic to L5(2) and corresponding to the roots 1, 2, 3, 4 , 1, 3, 4, 5 , 2, 4, 5, 6 , 3, 4, 5, 6 , 4, 5, 6, 7 and { } { } { } { } { } 5, 6, 7, 8 respectively. As before, for each of these groups, we need to choose an { } element of order 31, noted as x1,x2,x3,x4,x5,x6 respectively, as well as the extended centraliser CG⇤ (xi)foreach.

From Table 3.1.4 the centraliser of an element of 31ABC has centraliser 31 L (2). ⇥ 5 To find CG(x1) we use the extended Dynkin diagram to see that there is an L5(2) generated by the roots 6, 7, 8 and the longest root which commutes with P (1234) and { } therefore x1 and so we have CG(x1). We then find all other CG(xi)usingFindCentProc having picked the other xi to be of the appropriate conjugacy class in E8(2). To find an inverting involution for the xi we choose an element of order 5 in CG(xi)inE8(2)-class

5A and ﬁnd its centraliser in E8(2). We then ﬁnd the normaliser of in this centraliser and choose an inverting involution from there. This gives us all CG⇤ (xi)and in each there are 13888 inverting involutions for xi. We shall now describe the results for each parabolic that has to be checked. In this section we use the notation 84/64 to denote a subgroup of order 284 whose Frattini subgroup is of order 264, equivalently any other values in place of 84 and 64.

5.3.1 P(1234)