# Book:Robert A. Wilson/The Finite Simple Groups

## Robert A. Wilson: The Finite Simple Groups

Published $\text {2009}$, Springer-Verlag

ISBN 978-1-84800-987-5.

### Contents

1 Introduction
1.1 A brief history of simple groups
1.2 The Classification Theorem
1.3 Applications of the Classification Theorem
1.4 Remarks on the proof of the Classification Theorem
1.5 Prerequisites
1.6 Notation
1.7 How to read this book
2 The alternating groups
2.1 Introduction
2.2 Permutations
2.2.1 The alternating groups
2.2.2 Transitivity
2.2.3 Primitivity
2.2.4 Group actions
2.2.5 Maximal subgroups
2.2.6 Wreath products
2.3 Simplicity
2.3.1 Cycle types
2.3.2 Conjugacy classes in the alternating groups
2.3.3 The alternating groups are simple
2.4 Outer automorphisms
2.4.1 Automorphisms of alternating groups
2.4.2 The outer automorphism of $S_6$
2.5 Subgroups of $S_n$
2.5.1 Intransitive subgroups
2.5.2 Transitive imprimitive subgroups
2.5.3 Primitive wreath products
2.5.4 Affine subgroups
2.5.5 Subgroups of diagonal type
2.5.6 Almost simple groups
2.6 The O'Nan–Scott Theorem
2.6.1 General results
2.6.2 The proof of the O'Nan–Scott Theorem
2.7 Covering groups
2.7.1 The Schur multiplier
2.7.2 The double covers of $A_n$ and $S_n$
2.7.3 The triple cover of $A_6$
2.7.4 The triple cover of $A_7$
2.8 Coxeter groups
2.8.1 A presentation of $S_n$
2.8.2 Real reflection groups
2.8.3 Roots, root systems, and root lattices
2.8.4 Weyl groups
Exercises
3 The classical groups
3.1 Introduction
3.2 Finite fields
3.3 General linear groups
3.3.1 The orders of the linear groups
3.3.2 Simplicity of $\mathrm{PSL}_n(q)$
3.3.3 Subgroups of the linear groups
3.3.4 Outer automorphisms
3.3.5 The projective line and some exceptional isomorphisms
3.3.6 Covering groups
3.4 Bilinear, sesquilinear and quadratic forms
3.4.1 Definitions
3.4.2 Vectors and subspaces
3.4.3 Isometries and similarities
3.4.4 Classification of alternating bilinear forms
3.4.5 Classification of sesquilinear forms
3.4.6 Classification of symmetric bilinear forms
3.4.7 Classification of quadratic forms in characteristic $2$
3.4.8 Witt's Lemma
3.5 Symplectic groups
3.5.1 Symplectic transvections
3.5.2 Simplicity of $\mathrm{PS}_{\mathrm P_{2m}} (q)$
3.5.3 Subgroups of symplectic groups
3.5.4 Subspaces of a symplectic space
3.5.5 Covers and automorphisms
3.6 Unitary groups
3.6.1 Simplicity of unitary groups
3.6.2 Subgroups of unitary groups
3.6.3 Outer automorphisms
3.6.5 Exceptional behaviour
3.7 Orthogonal groups in odd characteristic
3.7.1 Determinants and spinor norms
3.7.2 Orders of orthogonal groups
3.7.3 Simplicity of $\mathrm P\Omega_n(q)$
3.7.4 Subgroups of orthogonal groups
3.7.5 Outer automorphisms
3.8 Orthogonal groups in characteristic $2$
3.8.1 The quasideterminant and the structure of the groups
3.8.2 Properties of orthogonal groups in characteristic $2$
3.9 Clifford algebras and spin groups
3.9.1 The Clifford algebra
3.9.2 The Clifford group and the spin group
3.9.3 The spin representation
3.10 Maximal subgroups of classical groups
3.10.1 Tensor products
3.10.2 Extraspecial groups
3.10.3 The Aschbacher–Dynkin theorem for linear groups
3.10.4 The Aschbacher–Dynkin theorem for classical groups
3.10.5 Tensor products of spaces with forms
3.10.6 Extending the field on spaces with forms
3.10.7 Restricting the field on spaces with forms
3.10.8 Maximal subgroups of symplectic groups
3.10.9 Maximal subgroups of unitary groups
3.10.10 Maximal subgroups of orthogonal groups
3.11 Generic isomorphisms
3.11.1 Low-dimensional orthogonal groups
3.11.2 The Klein correspondence
3.12 Exceptional covers and isomorphisms
3.12.1 Isomorphisms using the Klein correspondence
3.12.2 Covering groups of $\mathrm{PSU}_4(3)$
3.12.3 Covering groups of $\mathrm{PSL}_3(4)$
3.12.4 The exceptional Weyl groups
Exercises
4 The exceptional groups
4.1 Introduction
4.2 The Suzuki groups
4.2.1 Motivation and definition
4.2.2 Generators for $\mathrm{Sz}(q)$
4.2.3 Subgroups
4.2.4 Covers and automorphisms
4.3 Octonions and groups of type $G_2$
4.3.1 Quaternions
4.3.2 Octonions
4.3.3 The order of $G_2(q)$
4.3.4 Another basis for the octonions
4.3.5 The parabolic subgroups of $G_2(q)$
4.3.6 Other subgroups of $G_2(q)$
4.3.7 Simplicity of $G_2(q)$
4.3.8 The generalised hexagon
4.3.9 Automorphisms and covers
4.4 Integral octonions
4.4.1 Quaternions in characteristic $2$
4.4.2 Integral octonions
4.4.3 Octonions in characteristic $2$
4.4.4 The isomorphism between $G_2(2)$ and $\mathrm{PSU}_3(3)\colon 2$
4.5 The small Ree groups
4.5.1 The outer automorphism of $G_2(3)$
4.5.2 The Borel subgroup of $^2G_2(q)$
4.5.3 Other subgroups
4.5.4 The isomorphism $^2G_2(3) \simeq P\Gamma L_2(8)$
4.6 Twisted groups of type $^3D_4$
4.6.1 Twisted octonion algebras
4.6.2 The order of $^3D_4(q)$
4.6.3 Simplicity
4.6.4 The generalised hexagon
4.6.5 Maximal subgroups of $^3D_4(q)$
4.7 Triality
4.7.1 Isotopies
4.7.2 The triality automorphism of $\mathrm P\Omega^+_8 (q)$
4.7.3 The Klein correspondence revisited
4.8 Albert algebras and groups of type $F_4$
4.8.1 Jordan algebras
4.8.2 A cubic form
4.8.3 The automorphism groups of the Albert algebras
4.8.4 Another basis for the Albert algebra
4.8.5 The normaliser of a maximal torus
4.8.6 Parabolic subgroups of $F_4(q)$
4.8.7 Simplicity of $F_4(q)$
4.8.8 Primitive idempotents
4.8.9 Other subgroups of $F_4(q)$
4.8.10 Automorphisms and covers of $F_4(q)$
4.8.11 An integral Albert algebra
4.9 The large Ree groups
4.9.1 The outer automorphism of $F_4(2)$
4.9.2 Generators for the large Ree groups
4.9.3 Subgroups of the large Ree groups
4.9.4 Simplicity of the large Ree groups
4.10 Trilinear forms and groups of type $E_6$
4.10.1 The determinant
4.10.2 Dickson's construction
4.10.3 The normaliser of a maximal torus
4.10.4 Parabolic subgroups of $E_6(q)$
4.10.5 The rank $3$ action
4.10.6 Covers and automorphisms
4.11 Twisted groups of type $^2E_6$
4.12 Groups of type $E_7$ and $E_8$
4.12.1 Lie algebras
4.12.2 Subgroups of $E_8(q)$
4.12.3 $E_7(q)$
Exercises
5.1 Introduction
5.2 The large Mathieu groups
5.2.1 The hexacode
5.2.2 The binary Golay code
5.2.3 The group $M_{24}$
5.2.4 Uniqueness of the Steiner system $S(5, 8,24)$
5.2.5 Simplicity of $M_{24}$
5.2.6 Subgroups of $M_{24}$
5.2.7 A presentation of $M_{24}$
5.2.8 The group $M_{23}$
5.2.9 The group $M_{22}$
5.2.10 The double cover of $M_{22}$
5.3 The small Mathieu groups
5.3.1 The group $M_{12}$
5.3.2 The Steiner system $S(5, 6,12)$
5.3.3 Uniqueness of $S(5, 6,12)$
5.3.4 Simplicity of $M_{12}$
5.3.5 The ternary Golay code
5.3.6 The outer automorphism of $M_{12}$
5.3.7 Subgroups of $M_{12}$
5.3.8 The group $M_{11}$
5.4 The Leech lattice and the Conway group
5.4.1 The Leech lattice
5.4.2 The Conway group $\mathrm{Co}_1$
5.4.3 Simplicity of $\mathrm{Co}_1$
5.4.4 The small Conway groups
5.4.5 The Leech lattice modulo $2$
5.5 Sublattice groups
5.5.1 The Higman–Sims group $\mathrm{HS}$
5.5.2 The McLaughlin group $\mathrm{McL}$
5.5.3 The group $\mathrm{Co}_3$
5.5.4 The group $\mathrm{Co}_2$
5.6 The Suzuki chain
5.6.1 The Hall–Janko group $\mathrm J_2$
5.6.2 The icosians
5.6.3 The icosian Leech lattice
5.6.4 Properties of the Hall–Janko group
5.6.5 Identification with the Leech lattice
5.6.6 $\mathrm J_2$ as a permutation group
5.6.7 Subgroups of $\mathrm J_2$
5.6.8 The exceptional double cover of $G_2(4)$
5.6.9 The map onto $G_2(4)$
5.6.10 The complex Leech lattice
5.6.11 The Suzuki group
5.6.12 An octonion Leech lattice
5.7 The Fischer groups
5.7.1 A graph on $3510$ vertices
5.7.2 The group $\mathrm{Fi}_{22}$
5.7.3 Conway's description of $\mathrm{Fi}_{22}$
5.7.4 Covering groups of $\mathrm{Fi}_{22}$
5.7.5 Subgroups of $\mathrm{Fi}_{22}$
5.7.6 The group $\mathrm{Fi}_{23}$
5.7.7 Subgroups of $\mathrm{Fi}_{23}$
5.7.8 The group $\mathrm{Fi}_{24}$
5.7.9 Parker's loop
5.7.10 The triple cover of $\mathrm{Fi}'_{24}$
5.7.11 Subgroups of $\mathrm{Fi}_{24}$
5.8 The Monster and subgroups of the Monster
5.8.1 The Monster
5.8.2 The Griess algebra
5.8.3 $6$-transpositions
5.8.4 Monstralisers and other subgroups
5.8.5 The Y-group presentations
5.8.6 The Baby Monster
5.8.7 The Thompson group
5.8.9 The Held group
5.8.10 Ryba's algebra
5.9 Pariahs
5.9.1 The first Janko group $\mathrm J_1$
5.9.2 The third Janko group $\mathrm J_3$
5.9.3 The Rudvalis group
5.9.4 The O'Nan group
5.9.5 The Lyons group
5.9.6 The largest Janko group $\mathrm J_4$