Book:Robert A. Wilson/The Finite Simple Groups

Subject Matter

 * Group Theory

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
 * Further reading
 * 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
 * 3.5.6 The generalised quadrangle
 * 3.6 Unitary groups
 * 3.6.1 Simplicity of unitary groups
 * 3.6.2 Subgroups of unitary groups
 * 3.6.3 Outer automorphisms
 * 3.6.4 Generalised quadrangles
 * 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
 * Further reading
 * 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 $E_7(q)$
 * Further reading
 * Exercises


 * 5 The sporadic groups
 * 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
 * 5.8.8 The Harada–Norton 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 $\mathrm J_4$
 * Further reading
 * Exercises


 * References
 * Index