Some Calculations Related To The Monster Group by Richard William Barraclough A thesis submitted to The University of Birmingham for the degree of Doctor of Philosophy School of Mathematics and Statistics The University of Birmingham September 2005 Synopsis This thesis describes some computer calculations relating to the Monster group. In Chapter 2 we describe the computer construction of the Monster carried out by Linton et al. in 1997, and again (over a different field) in 2000 by Wilson. We modify the latter programs, considerably in- creasing the speed and thus the feasibility of using them for practical computations. We make use of our modified programs in Chapter 3 to determine conjugacy class representatives for the Monster group, up to algebraic conjugacy. This completes the work of Wilson et al. on conjugacy class representatives for sporadic simple groups. Chapter 4 describes the method of “Fischer matrices” for computing character tables of groups with an elementary Abelian or extraspecial normal subgroup. Fischer’s own rather cryptic paper on the subject has inspired many more expository papers by Moori et al. and our proof of Theorem 12 adds to this revised exposition. The final section of the chapter describes a GAP function to compute the character . table of a group of shape (2 × 2 G):2 and gives (2 × 2 Fi22):2 6 Fi24 as an example. Chapter 5 finds most of the irreducible characters of the three-fold cover of a maximal subgroup of the Monster, N(3B). We construct various representations which we first of all use to determine the conjugacy classes, power maps, and quotient map onto N(3B). Observations in Chapter 3 aid us to find 463 of the 533 irreducible characters, but unfortunately we cannot find the remaining characters in the tensor algebra of our 463 irreducibles. As part of this work we also determine conjugacy class representatives for 6.Suz, including the algebraically conjugate classes. Chapter 6 works towards a classification of the “nets” of the Monster. These are combinatorial structures defined by Norton to encode information about subgroups of the Monster in a way sympathetic with his generalised Moonshine conjectures. We witness all nets centralised by an element of prime order at least 5, and can determine which of these could be conjugate in the Monster. The remaining cases are considerably harder, and our programs will not scale up. Instead we describe our strategy to extend our work to a complete classification. Acknowledgements I would like to thank my supervisor Professor R. A. Wilson for his help and guidance. I am grateful to Prof. R. T. Curtis, Dr. P. E. Holmes, Dr. P. J. Flavell, Dr. C. W. Parker, and in particular Dr. J. N. Bray, for their help and many useful conversations. I would also like to acknowledge other members past and present of the department who contributed to a very pleasant working environment. For financial assistance I acknowledge the EPSRC. Contributions via The London Mathematical Society, The Royal Society, The University of Birmingham, and other academic institutions have been noteworthy. Contents 1 Introduction & Motivation 1 2 A Computer Construction Of The Monster 7 2.1 Computer Construction Of Groups . 7 2.1.1 The Standard Basis Algorithm . 7 2.1.2 Subgroups H and K that intersect in a group L ..................... 8 2.2 Constructing The Monster Group . 9 2.2.1 Representing 31+12.2.Suz ................................. 9 2.2.2 Another Construction Over F7 .............................. 12 2.2.3 The Generators . 12 2.3 Programming The Construction . 13 2.3.1 Monster Operations (mop) . 13 2.3.2 Grease . 15 2.3.3 Driver Programs . 15 2.3.4 Other Modifications . 18 2.3.5 Experimental Versions And Ideas . 19 3 Conjugacy Class Representatives For The Monster Group 20 3.1 Distinguishing Conjugacy Classes . 20 3.2 Searching For Conjugacy Class Representatives . 21 3.3 Word Generation Strategies . 22 3.3.1 Example: Finding An Element In Class 24C ....................... 22 3.4 Elements of Order 27 . 23 3.5 Results . 24 3.6 Monster Element Database . 30 4 Fischer Matrices And The Character Tables Of Groups Of Shape 22.G:2 31 4.1 Some Clifford Theory . 32 4.2 Conjugacy Classes . 34 4.3 The Fischer Matrices . 35 4.4 Coset Analysis And Properties Of Fischer Matrices . 38 4.5 Extending G to G:2 (or vice versa)................................. 39 4.6 The Character Table Of 22.G:2................................... 41 2. 4.7 The Character Table Of 2 Fi22:2.................................. 42 4.7.1 Conjugacy Classes and Fischer Matrices . 43 4.7.2 From Class Functions To Characters . 47 4.7.3 Power Maps . 47 4.8 A GAP Function For The General Case . 48 5 The Character Tables Of A Maximal Subgroup Of The Monster, And Some Related Groups 52 5.1 The Conjugacy Classes Of 6.Suz:2 . 53 5.1.1 A Representation Of 312:6.Suz:2 . 55 5.1.2 A Representation Of 31+12:6.Suz:2 . 56 5.1.3 Conjugacy Classes . 58 5.2 The Quotient 31+12.2.Suz:2 and Power Maps . 60 5.2.1 The Quotient And Fusion To The Monster . 60 5.2.2 Power Maps . 61 5.2.3 The First 174 Characters . 62 5.3 The 729-dimensional representations of 31+12:6.Suz:2 . 62 5.3.1 A 729-dimensional module for 31+12 ........................... 63 5.3.2 Extending To 31+12:2.Suz . 64 5.3.3 Finding Standard Generators . 65 5.3.4 Adjoining The Automorphism: A 1458-dimensional Representation For 31+12:2.Suz:2 66 5.3.5 Finding The Character Values . 67 5.4 Induction From 31+12:6.Suz .................................... 68 5.5 Induction From The Index 65520 Inertia Group . 69 5.5.1 A Permutation Representation Of 312:6.Suz:2 . 69 12 5.5.2 The Group 3 :(2 × (3 × U5(2)):2) And A Quotient Of It . 70 5.5.3 Representations of S3 × (3 × U5(2)):2) . 71 5.5.4 Some Representations Of U5(2):2 . 73 5.5.5 The Induction . 73 5.6 Conclusions . 75 6 Nets 77 6.1 Definitions And Terminology . 77 6.2 Moonshine . 82 6.2.1 The Original Moonshine Conjectures . 82 6.2.2 Generalised Moonshine And The Link With Nets . 83 6.2.3 Monstralisers . 84 6.3 Finding Nets . 84 6.3.1 Finding Nets Centralised By An Element Of Prime Order At Least 7 . 85 6.3.2 Net Functions: Data Structures And Supporting Functions . 85 6.3.3 Simple Net Finding Program . 88 6.3.4 Results for p > 11...................................... 89 6.3.5 Results for p = 7 ...................................... 93 6.4 A Presentation For The Stabiliser Of A Flag . 97 6.5 Nets Centralised By An Element Of Class 5B . 101 6.6 Nets Centralised By An Element Of Class 5A . 103 6.6.1 Finding The Conjugacy Classes Of Triples Of Involutions . 103 6.6.2 Finding The Conjugacy Classes Of Nets . 104 6.7 Further Work . 105 6.7.1 Nets Centralised By An Element Of Class 3B . 106 6.7.2 Nets Centralised By An Element Of Class 3A . 107 6.7.3 Nets That Generate The Monster . 107 A The CD 109 A.1 A Computer Construction Of The Monster . 109 A.2 Conjugacy Class Representatives In The Monster Group . 110 A.3 Fischer Matrices . 112 A.4 The Character Tables Of A Maximal Subgroup Of The Monster, And Some Related Groups 112 A.5 Nets . 115 B Conjugacy Class Identification 116 List of References 119 List of Figures 1 Some groups . 35 . 2 Conjugacy classes of 2 Fi22. .................................... 44 . 3 The character table of (2 × 2 A6) 23. (Computed in Magma.) . 50 . 4 The character table of (4 ◦ 2 A6) 23. (Stored in GAP.) . 51 5 Anet.................................................. 79 6 An “exploded” net. 98 7 Choosing a spanning tree for use with the Reidemeister-Schreier algorithm. 99 8 Choosing a spanning tree from a net. 100 Chapter 1 Introduction & Motivation group G with a normal subgroup N and quotient isomorphic to Q, i.e., G = N.Q, may be understood through an investigation of Q and the action of Q on N. However, if G is simple A then it has no proper non-trivial normal subgroups and so this reduction is not possible. We therefore find ourselves interested in simple groups, the building blocks of all groups. Let G be a finite simple group. According to the celebrated Classification Of Finite Simple Groups [14] G is one of the following groups: (i) A cyclic group of prime order, Cp. (ii) An alternating group An for n > 5. (iii) A Chevalley group. For each of the root systems Xn ∈ {An, Bn, Cn, Dn, F4, G2, E6, E7, E8} one can obtain a finite group Xn(q), most of which are simple. This approach provides a uniform construc- tion of the classical groups, as well as a number of new infinite families. • The classical groups are recovered as: ∼ – An(q) = Ln+1(q) for n > 1. 2 ∼ – An(q) = Un+1(q) for n > 2. ∼ – Bn(q) = O2n+1(q) for n > 2. ∼ – Cn(q) = S2n(q) for n > 3. ∼ + – Dn(q) = O2n(q).