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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 Copyright Statement 7 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 intel- lectual 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 rele- vant Thesis restriction declarations deposited in the University Library, The Univer- sity Library’s regulations (see http://www.manchester.ac.uk/library/aboutus/regul- ations) 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 sub- groups 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 the- orem, initially proved independently by O’Nan and Scott; however, due to the impor- tance 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 com- putational 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].
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