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The Finite Simple Groups

:	Robert Wilson
:	The Finite Simple Groups
:	Springer Verlag London Limited
:	9781848009882
:	1
:	CHF 47.60
:

:	Naturwissenschaft
:	English

:	310
:	DRM
:	PC/MAC/eReader/Tablet
:	PDF

The finite simple groups are the building blocks from which all the finite groups are made and as such they are objects of fundamental importance throughout mathematics. The classification of the finite simple groups was one of the great mathematical achievements of the twentieth century, yet these groups remain difficult to study which hinders applications of the classification.

This textbook brings the finite simple groups to life by giving concrete constructions of most of them, sufficient to illuminate their structure and permit real calculations both in the groups themselves and in the underlying geometrical or algebraic structures. This is the first time that all the finite simple groups have been treated together in this way and the book points out their connections, for example between exceptional behaviour of generic groups and the existence of sporadic groups, and discusses a number of new approaches to some of the groups. Many exercises of varying difficulty are provided.

	Preface	5
	Contents	9
	Introduction	16
	A brief history of simple groups	16
	The Classification Theorem	18
	Applications of the Classification Theorem	19
	Remarks on the proof of the Classification Theorem	20
	Prerequisites	21
	Notation	24
	How to read this book	25
	The alternating groups	26
	Introduction	26
	Permutations	26
	The alternating groups	27
	Transitivity	28
	Primitivity	28
	Group actions	29
	Maximal subgroups	29
	Wreath products	30
	Simplicity	31
	Cycle types	31
	Conjugacy classes in the alternating groups	31
	The alternating groups are simple	32
	Outer automorphisms	33
	Automorphisms of alternating groups	33
	The outer automorphism of S6	34
	Subgroups of Sn	34
	Intransitive subgroups	35
	Transitive imprimitive subgroups	35
	Primitive wreath products	36
	Affine subgroups	36
	Subgroups of diagonal type	37
	Almost simple groups	37
	The O'Nan--Scott Theorem	38
	General results	39
	The proof of the O'Nan--Scott Theorem	41
	Covering groups	42
	The Schur multiplier	42
	The double covers of An and Sn	43
	The triple cover of A6	44
	The triple cover of A7	45
	Coxeter groups	46
	A presentation of Sn	46
	Real reflection groups	47
	Roots, root systems, and root lattices	48
	Weyl groups	49
	Further reading	50
	Exercises	50
	The classical groups	55
	Introduction	55
	Finite fields	56
	General linear groups	57
	The orders of the linear groups	58
	Simplicity of PSLn(q)	59
	Subgroups of the linear groups	60
	Outer automorphisms	62
	The projective line and some exceptional isomorphisms	64
	Covering groups	67
	Bilinear, sesquilinear and quadratic forms	67
	Definitions	68
	Vectors and subspaces	69
	Isometries and similarities	70
	Classification of alternating bilinear forms	70
	Classification of sesquilinear forms	71
	Classification of symmetric bilinear forms	71
	Classification of quadratic forms in characteristic 2	72
	Witt's Lemma	73
	Symplectic groups	74
	Symplectic transvections	75
	Simplicity of PSp2m(q)	75
	Subgroups of symplectic groups	76
	Subspaces of a symplectic space	77
	Covers and automorphisms	78
	The generalised quadrangle	78
	Unitary groups	79
	Simplicity of unitary groups	80
	Subgroups of unitary groups	81
	Outer automorphisms	82
	Generalised quadrangles	82
	Exceptional behaviour	83
	Orthogonal groups in odd characteristic	83
	Determinants and spinor norms	84
	Orders of orthogonal groups	85
	Simplicity of Pn(q)	86
	Subgroups of orthogonal groups	88
	Outer automorphisms	89
	Orthogonal groups in characteristic 2	90
	The quasideterminant and the structure of the groups	90
	Properties of orthogonal groups in characteristic 2	91
	Clifford algebras and spin groups	92
	The Clifford algebra	93
	The Clifford group and the spin group	93
	The spin representation	94
	Maximal subgroups of classical groups	95
	Tensor products	96
	Extraspecial groups	97
	The Aschbacher--Dynkin theorem for linear groups	99
	The Aschbacher--Dynkin theorem for classical groups	100
	Tensor products of spaces with forms	101
	Extending the field on spaces with forms	103
	Restricting the field on spaces with forms	104
	Maximal subgroups of symplectic groups	106
	Maximal subgroups of unitary groups	107
	Maximal subgroups of orthogonal groups	108
	Generic isomorphisms	110
	Low-dimensional orthogonal groups	110
	The Klein correspondence	111
	Exceptional covers and isomorphisms	113
	Isomorphisms using the Klein correspondence	113
	Covering groups of PSU4(3)	114
	Covering groups of PSL3(4)	115
	The exceptional Weyl groups	117
	Further reading	119
	Exercises	120
	The exceptional groups	124
	Introduction	124
	The Suzuki groups	126
	Motivation and definition	126
	Generators for Sz(q)	128
	Subgroups	130
	Covers and automorphisms	131
	Octonions and groups of type G2	131
	Quatern