GRAPHS ASSOCIATED WITH SPORADIC GROUP GEOMETRIES, AND THE SEMISIMPLE ELEMENTS OF E8(2) A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Engineering and Physical Sciences 2016 Jamie Steven Phillips School of Mathematics Contents Abstract 10 Declaration 11 Copyright Statement 12 Acknowledgements 13 1 Introduction 14 2 Background Material 19 2.1 General Notation . 19 2.2 Finite Simple Groups and ATLAS Notation . 20 2.2.1 Finite Simple Groups . 20 2.2.2 ATLAS Notation . 21 2.3 p-Groups . 22 2.4 Buildings . 24 2.5 Sporadic Group Geometries . 27 2.5.1 Definitions and Notation . 27 2.5.2 Parabolic Subgroups and 2-Local Geometries . 29 3 Plane-Line Collinearity Graphs 33 3.1 Literature Review . 34 3.1.1 Point-Line Collinearity Graphs . 34 3.1.2 Plane-Line Collinearity Graphs . 38 3.2 Plane-Line Graph for M23 ............................. 40 3.2.1 The Minimal 2-local Geometry . 41 3.2.2 The Graph . 42 3.2.3 The Computer Files . 45 2 3.3 Plane-Line Graph for J4 .............................. 46 3.3.1 The Maximal 2-local Geometry . 47 3.3.2 The Graph . 48 3.3.3 The Computer Files . 52 3.4 Plane-Line Graph for F i22 ............................. 53 3.4.1 The Minimal 2-Local Geometry . 54 3.4.2 The Graph . 55 3.4.3 The Computer Files . 57 3.5 Plane-Line Graph for F i23 ............................. 58 3.5.1 The Minimal 2-Local Geometry . 58 3.5.2 The Graph . 59 3.5.3 The Computer Files . 67 3.6 Plane-Line Graph for He ............................. 68 3.6.1 The Minimal 2-local Geometry . 69 3.6.2 The Graph . 70 3.6.3 The Computer Files . 74 3.7 Plane-Line Graph for Co3 ............................. 75 3.7.1 The Maximal 2-Local Geometry . 76 3.7.2 The Graph . 77 3.7.3 The Computer Files . 86 3.8 Plane-Line Graph for Co2 ............................. 87 3.8.1 The Maximal 2-Local Geometry . 87 3.8.2 The Graph . 88 3.8.3 The Computer Files . 89 3.9 Remaining Cases . 89 4 Graphs Associated with McL 91 4.1 The First Minimal 2-Local Geometry for McL ................. 92 4.1.1 The Graphs . 92 4.2 The Second Minimal 2-Local Geometry for McL ................ 97 4.2.1 The Graphs . 97 4.3 The Third and Fourth Minimal 2-Local Geometries . 101 4.4 Computer Files . 103 3 5 Commuting Graphs for B 104 5.1 Introduction . 104 5.2 Permutation Rank for (B; 2C)........................... 107 5.3 Graph for (B; 2D)................................. 108 6 The Semisimple Elements of E8(2) 114 6.1 Introduction . 114 6.2 Preliminary Results . 121 6.3 Prime Order Elements . 126 6.4 Composite Order Elements . 132 6.4.1 Elements of order 5m ........................... 132 6.4.2 Elements of order 7m ........................... 137 6.4.3 Elements of order 11m ........................... 139 6.4.4 Elements of order 13m ........................... 140 6.4.5 Elements of order 17m ........................... 141 6.4.6 Remaining Cases . 142 6.5 Fixed-Point Spaces and Powering Up Maps . 145 6.5.1 Fixed-Point Spaces . 145 6.5.2 Powering Up Maps . 146 References 148 Word count 33447 4 List of Tables 3.1 The Collapsed Adjacency Matrix for the Plane-Line Collinearity Graph of M23 43 3.2 The Gh-orbits of the Plane-Line Collinearity Graph for M23 .......... 44 3.3 The Plane-Line Distribution in the Collinearity Graph for M23 ........ 45 3.4 The Collapsed Adjacency Matrix for the Plane-Line Collinearity Graph of J4 50 3.5 The Gh-orbits of the Plane-Line Collinearity Graph for J4 ........... 51 3.6 The Collapsed Adjacency Matrix for the Plane-Line Collinearity Graph of F i22 56 3.7 The Gh-orbits of the Plane-Line Collinearity Graph for F i22 .......... 57 3.8 The Gh-orbits of the Plane-Line Collinearity Graph for F i23 .......... 60 3.9 The Gh-orbits of the Plane-Line Collinearity Graph for He .......... 71 3.10 The Collapsed Adjacency Matrix for the Plane-Line Collinearity Graph of He 72 3.11 The Gh-orbits of the Plane-Line Collinearity Graph for Co3 .......... 78 3.12 The Collapsed Adjacency Matrix for the Plane-Line Collinearity Graph of Co3 80 3.13 The Gh-orbits of the Plane-Line Collinearity Graph for Co2 .......... 88 4.1 The Collapsed Adjacency Matrix for a Line-Point Collinearity Graph of McL 94 4.2 The Gl-orbits of a Line-Point Collinearity Graph for McL ........... 94 4.3 The Collapsed Adjacency Matrix for a Line-Plane Collinearity Graph of McL 96 4.4 The Collapsed Adjacency Matrix for a Plane-Line Collinearity Graph of McL 99 5.1 The Class Structure Constants for C(B; 2D).................... 109 5.2 The Location of some XC in the Commuting Involution Graph C(B; 2D)... 113 6.1 The Conjugacy Classes of Semisimple Elements in E8(2) . 117 6.2 Lower Bounds on the Number of Conjugacy Classes in E8(2) . 123 6.3 The Conjugacy Classes of Elements of Order 3 in E8(2) . 126 6.4 The Conjugacy Classes of Elements of Order 5 in E8(2) . 127 6.5 The Conjugacy Classes of Elements of Order 7 in E8(2) . 127 6.6 The Conjugacy Class of Elements of Order 11 in E8(2) . 128 5 6.7 The Conjugacy Classes of Elements of Order 13 in E8(2) . 128 6.8 The Conjugacy Classes of Elements of Order 17 in E8(2) . 129 6.9 The Conjugacy Class of Elements of Order 19 in E8(2) . 130 6.10 The Conjugacy Classes of Elements of Order 31 in E8(2) . 130 6.11 The Conjugacy Classes of Elements of Order 41 in E8(2) . 131 6.12 The Conjugacy Classes of Elements of Order 43 in E8(2) . 131 6.13 The Conjugacy Classes of Elements of Order 73 in E8(2) . 131 6.14 The Conjugacy Classes of Elements of Order 127 in E8(2) . 132 6.15 The Conjugacy Classes of Elements of Orders 151, 241 and 331 in E8(2) . 132 6.16 The Conjugacy Classes of Elements of Order 15 in E8(2) . 133 6.17 The Conjugacy Classes of Elements of Order 35 in E8(2) . 133 6.18 The Conjugacy Classes of Elements of Order 45 in E8(2) . 134 6.19 The Conjugacy Classes of Elements of Order 55 in E8(2) . 134 6.20 The Conjugacy Classes of Elements of Order 65 in E8(2) . 135 6.21 The Conjugacy Classes of Elements of Order 85 in E8(2) . 135 6.22 The Conjugacy Classes of Elements of Order 105 in E8(2) . 135 6.23 The Conjugacy Classes of Elements of Order 155 in E8(2) . 136 6.24 The Conjugacy Classes of Elements of Order 165 in E8(2) . 136 6.25 The Conjugacy Classes of Elements of Order 195 in E8(2) . 136 6.26 The Conjugacy Classes of Elements of Order 255 in E8(2) . 137 6.27 The Conjugacy Classes of Elements of Order 315 in E8(2) . 137 6.28 The Conjugacy Classes of Elements of Order 21 in E8(2) . 138 6.29 The Conjugacy Classes of Elements of Order 91 in E8(2) . 138 6.30 The Conjugacy Classes of Elements of Order 119 in E8(2) . 138 6.31 The Conjugacy Classes of Elements of Order 217 in E8(2) . 139 6.32 The Conjugacy Classes of Elements of Order 511 in E8(2) . 139 6.33 The Conjugacy Classes of Elements of Order 33 in E8(2) . 139 6.34 The Conjugacy Classes of Elements of Order 99 in E8(2) . 140 6.35 The Conjugacy Classes of Elements of Order 39 in E8(2) . 140 6.36 The Conjugacy Classes of Elements of Order 117 in E8(2) . 141 6.37 The Conjugacy Classes of Elements of Order 273 in E8(2) . 141 6.38 The Conjugacy Classes of Elements of Order 51 in E8(2) . 141 6.39 The Conjugacy Classes of Elements of Order 153 in E8(2) . 142 6.40 The Conjugacy Classes of Elements of Order 93 in E8(2) . 142 6 6.41 The Conjugacy Classes of Elements of Order 129 in E8(2) . 143 6.42 The Conjugacy Classes of Elements of Order 219 in E8(2) . 143 6.43 The Conjugacy Classes of Elements of Order 381 in E8(2) . 143 6.44 The Conjugacy Classes of Elements of Order 9 in E8(2) . 144 6.45 The Conjugacy Classes of Elements of Order 63 in E8(2) . 144 6.46 The Conjugacy Classes of Elements of Orders 57 and 171 in E8(2) . 145 7 List of Figures 1.1 A Plane-Line Collinearity Graph for McL .................... 16 2.1 The Finite Projective Plane of Order 2 . 28 2.2 The Maximal 2-local Geometry for J4 ...................... 32 3.1 The Minimal 2-local Geometry for M23 ...................... 41 3.2 A Minimal 2-local Geometry for Alt(7) . 41 3.3 The Plane-Line Collinearity Graph of M23 .................... 42 3.4 The Maximal 2-local Geometry for J4 ...................... 47 3.5 The Plane-Line Collinearity Graph of J4 ..................... 49 3.6 The Minimal 2-local Geometry for F i22 ..................... 54 3.7 The Plane-Line Collinearity Graph of F i22 .................... 56 3.8 The Minimal 2-local Geometry for F i23 ..................... 58 3.9 The Hyperplane-Plane Collinearity Graph for F i23 ..............