The theory of groups

Title page......Page 1
Copyright page......Page 2
Dedication......Page 3
Preface......Page 5
CONTENTS......Page 7
1.1 Algebraic Laws......Page 13
1.2 Mappings......Page 14
1.3 Definitions for Groups and Some Related Systems......Page 16
1.4 Subgroups, Isomorphisms, Homomorphisms......Page 19
1.5 Cosets. Theorem of Lagrange. Cyclic Groups. Indices......Page 22
1.6 Conjugates and Classes......Page 25
1.7 Double Cosets......Page 26
1.8 Remarks on Infinite Groups......Page 27
1.9 Examples of Groups......Page 31
2.1 Normal Subgroups......Page 38
2.3 Factor Groups......Page 39
2.4 Operators......Page 41
2.5 Direct Products and Cartesian Products......Page 44
3.1 Definition of Abelian Group. Cyclic Groups......Page 47
3.2 Some Structure Theorems for Abelian Groups......Page 48
3.3 Finite Abelian Groups. Invariants......Page 52
4.1 Falsity of the Converse of the Theorem of Lagrange......Page 55
4.2 The Three Sylow Theorems......Page 56
4.3 Finite $p$-Groups......Page 59
4.4 Groups of Orders $p$, $p^2$, $pq$, $p^3$......Page 61
5.1 Cycles......Page 65
5.2 Transitivity......Page 67
5.3 Representations of a Group by Permutations......Page 68
5.4 The Alternating Group $A_n$......Page 71
5.5 Intransitive Groups. Subdirect Products......Page 75
5.6 Primitive Groups......Page 76
5.7 Multiply Transitive Groups......Page 80
5.8 On a Theorem of Jordan......Page 84
5.9 The Wreath Product. Sylow Subgroups of Symmetric Groups......Page 93
6.2 Automorphisms of Groups. Inner Automorphisms......Page 96
6.3 The Holomorph of a Group......Page 98
6.4 Complete Groups......Page 99
6.5 Normal or Semi-direct Products......Page 100
7.1 Definition of Free Group......Page 103
7.2 Subgroups of Free Groups. The Schreier Method......Page 106
7.3 Free Generators of Subgroups of Free Groups. The Nielsen Method......Page 118
8.1 Partially Ordered Sets......Page 127
8.2 Lattices......Page 128
8.3 Modular and Semi-modular Lattices......Page 129
8.4 Principal Series and Composition Series......Page 135
8.5 Direct Decompositions......Page 139
8.6 Composition Series in Groups......Page 143
9.1 A Theorem of Frobenius......Page 148
9.2 Solvable Groups......Page 150
9.3 Extended Sylow Theorems in Solvable Groups......Page 153
9.4 Further Results on Solvable Groups......Page 157
10.2 The Lower and Upper Central Series......Page 161
10.3 Theory of Nilpotent Groups......Page 165
10.4 The Frattini Subgroup of a Group......Page 168
10.5 Supersolvable Groups......Page 170
11.1 The Collecting Process......Page 177
11.2 The Witt Formulae. The Basis Theorem......Page 180
12.2 The Burnside Basis Theorem. Automorphisms of $p$-Groups......Page 188
12.3 The Collection Formula......Page 190
12.4 Regular $p$-Groups......Page 195
12.5 Some Special $p$-Groups. Hamiltonian Groups......Page 199
13.1 Additive Groups. Groups Modulo One......Page 205
13.2 Characters of Abelian Groups. Duality of Abelian Groups......Page 206
13.3 Divisible Groups......Page 209
13.4 Pure Subgroups......Page 210
13.5 General Remarks......Page 211
14.1 Monomial Permutations......Page 212
14.2 The Transfer......Page 213
14.3 A Theorem of Burnside......Page 215
14.4 Theorems of P. Hall, Grün, and Wielandt......Page 216
15.1 Composition of Normal Subgroup and Factor Group......Page 230
15.2 Central Extensions......Page 234
15.3 Cyclic Extensions......Page 236
15.4 Defining Relations and Extensions......Page 238
15.5 Group Rings and Central Extensions......Page 240
15.6 Double Modules......Page 247
15.7 Cochains, Coboundaries, and Cohomology Groups......Page 248
15.8 Applications of Cohomology to Extension Theory......Page 252
16.2 Matrix Representation. Characters......Page 259
16.3 The Theorem of Complete Reducibility......Page 263
16.4 Semi-simple Group Rings and Ordinary Representations......Page 267
16.5 Absolutely Irreducible Representations. Structure of Simple Rings......Page 274
16.6 Relations on Ordinary Characters......Page 279
16.7 Imprimitive Representations......Page 293
16.8 Some Applications of the Theory of Characters......Page 297
16.9 Unitary and Orthogonal Representations......Page 306
16.10 Some Examples of Group Representation......Page 310
17.1 Definition of Free Product......Page 323
17.2 Amalgamated Products......Page 324
17.3 The Theorem of Kurosch......Page 327
18.2 The Burnside Problem for $n=2$ and $n=3$......Page 332
18.3 Finiteness of $B(4,r)$......Page 336
18.4 The Restricted Burnside problem. Theorems of P. Hall and G. Higman. Finiteness of $B(6,r)$......Page 337
19.1 General Properties......Page 351
19.2 Locally Cyclic Groups and Distributive Lattices......Page 352
19.3 The Theorem of Iwasawa......Page 354
20.1 Axioms......Page 358
20.2 Collineations and the Theorem of Desargues......Page 360
20.3 Introduction of Coordinates......Page 365
20.4 Veblen-Wedderburn Systems. Hall Systems......Page 368
20.5 Moufang and Desarguesian Planes......Page 378
20.6 The Theorem of Wedderburn and the Artin-Zorn Theorem......Page 387
20.7 Doubly Transitive Groups and Near-Fields......Page 394
20.8 Finite Planes. The Bruck-Ryser Theorem......Page 404
20.9 Collineations in Finite Planes......Page 410
BIBLIOGRAPHY......Page 433
INDEX......Page 441
INDEX OF SPECIAL SYMBOLS......Page 445