The Theory of Groups

Title Page......Page 1
Copyright Page......Page 3
Dedication......Page 5
Preface......Page 7
Contents......Page 9
1.1. Algebraic Laws......Page 14
1.2. Mappings......Page 15
1.3. Definitions for Groups and Some Related Systems......Page 17
1.4. Subgroups, Isomorphisms, Homomorphisms......Page 20
1.5. Cosets. Theorem of Lagrange. Cyclic groups. Indices......Page 23
1.6. Conjugates and Classes......Page 26
1.7. Double Cosets......Page 27
1.8. Remarks on Inflnite Groups......Page 28
1.9. Examples of Groups......Page 32
Exercises......Page 37
2.1. Normal Subgroups......Page 39
2.3. Factor Groups......Page 40
2.4. Operators......Page 42
2.5. Direct Products and Cartesian Products......Page 45
Exercises......Page 47
3.1. Deflnition of Abelian Group. Cyclic Groups......Page 48
3.2. Some Structure Theorems for Abelian Groups......Page 49
3.3. Finite Abelian Groups. Invariants......Page 53
Exercises......Page 55
4.1. Falsity of the Converse of the Theorem of Lagrange......Page 56
4.2. The Three Sylow Theorems......Page 57
4.3. Finite p-Groups......Page 60
4.4. Groups of Orders p, p², pq, p³......Page 62
Exercises......Page 65
5.1. Cycles......Page 66
5.2. Transitivity......Page 68
5.3. Representations of a Group by Permutations......Page 69
5.4. The Alternating Group A n......Page 72
5.5. Intransitive Groups. Subdirect Products......Page 76
5.6. Primitive Groups......Page 77
5.7. Multiply Transitive Groups......Page 81
5.8. On a Theorem of Jordan......Page 85
5.9. The Wreath Product. Sylow Subgroups of Symmetric Groups......Page 94
Exercises......Page 96
6.2. Automorphisms of Groups. Inner Automorphisms......Page 97
6.3. The Holomorph of a Group......Page 99
6.4. Complete Groups......Page 100
6.5. Normal or Semi-direct Products......Page 101
Exercises......Page 103
7.1. Definition of Free Group......Page 104
7.2. Subgroups of Free Groups. The Schreier Method......Page 107
7.3. Free Generators of Subgroups of Free Groups. The Nielsen Method......Page 119
Exercises......Page 127
8.1. Partially Ordered Sets......Page 128
8.2. Lattices......Page 129
8.3. Modular and Semi-modular Lattices......Page 130
8.4. Principal Series and Composition Series......Page 136
8.5. Direct Decompositions......Page 140
8.6. Composition Series in Groups......Page 144
Exercises......Page 147
9.1. A Theorem of Frobenius......Page 149
9.2. Solvable Groups......Page 151
9.3. Extended Sylow Theorems in Solvable Groups......Page 154
9.4. Further Results on Solvable Groups......Page 158
Exercises......Page 161
10.2. The Lower and Upper Central Series......Page 162
10.3. Theory of Nilpotent Groups......Page 166
10.4. The Frattini Subgroup of a Group......Page 169
10.5. Supersolvable Groups......Page 171
Exercises......Page 176
11.1. The Collecting Process......Page 178
11.2. The Witt Formulae. The Basis Theorem......Page 181
12.2. The Burnside Basis Theorem. Automorphisms of p-Groups......Page 189
12.3. The Collection Formula......Page 191
12.4. Regular p-Groups......Page 196
12.5. Some Special p-Groups. Hamiltonian Groups......Page 200
13.1. Additive Groups. Groups Modulo One......Page 206
13.2. Characters of Abelian Groups. Duality of Abelian Groups......Page 207
13.3. Divisible Groups......Page 210
13.4. Pure Subgroups......Page 211
13.5. General Remarks......Page 212
14.1. Monomial Permutations......Page 213
14.2. The Transfer......Page 214
14.3. A Theorem of Burnside......Page 216
14.4. Theorems of P. Hall, Grün, and Wielandt......Page 217
15.1. Composition of Normal Subgroup and Factor Group......Page 231
15.2. Central Extensions......Page 235
15.3. Cyclic Extensions......Page 237
15.4. Defining Relations and Extensions......Page 239
15.5. Group Rings and Central Extensions......Page 241
15.6. Double Modules......Page 248
15.7. Cochains, Coboundaries, and Cohomology Groups......Page 249
15.8. Applications of Cohomology to Extension Theory......Page 253
16.2. Matrix Representation. Characters......Page 260
16.3. The Theorem of Complete Reducibility......Page 264
16.4. Semi-simple Group Rings and Ordinary Representations......Page 268
16.5. Absolutely Irreducible Representations. Structure of Simple Rings......Page 275
16.6. Relations on Ordinary Characters......Page 280
16.7. Imprimitive Representations......Page 294
16.8. Some Applications of the Theory of Characters......Page 298
16.9. Unitary and Orthogonal Representations......Page 307
16.10. Some Examples of Group Representation......Page 311
17.1. Definition of Free Product......Page 324
17.2. Amalgamated Products......Page 325
17.3. The Theorem of Kurosch......Page 328
18.2. The Burnside Problem for n = 2 and n = 3......Page 333
18.3. Finiteness of B(4, r)......Page 337
18.4. The Restricted Burnside Problem. Theorems of P. Hall and G. Higman. Finiteness of B(6, r)......Page 338
19.1. General Properties......Page 352
19.2. Locally Cyclic Groups and Distributive Lattices......Page 353
19.3. The Theorem of Iwasawa......Page 355
20.1. Axioms......Page 359
20.2. Collineations and the Theorem of Desargues......Page 361
20.3. Introduction of Coordinates......Page 366
20.4. Veblen–Wedderburn Systems. Hall Systems......Page 369
20.5. Moufang and Desarguesian Planes......Page 379
20.6. The Theorem of Wedderburn and the Artin–Zorn Theorem......Page 388
20.7. Doubly Transitive Groups end Near-Fields......Page 395
20.8. Finite Planes. The Bruck–Ryser Theorem......Page 405
20.9. Collineations in Finite Planes......Page 411
Bibliography......Page 434
Index......Page 442
Index of Special Symbols......Page 446