This book is an excellent second book after the one from J Rotman (An introduction to the theory of groups).
Self studying with the 2 books demands a lot of work but can be achieved by someone motivated. When a concept is not understood with one of the books, the explanation read in the second really help.
This book is not for a beginner, but is perfect for someone with a real mathematical background. The concepts are organised as can be the mathematical courses in universities, concise, succint and demanding.
Author(s): Derek J.S. Robinson
Series: Graduate Texts in Mathematics
Edition: 2nd
Publisher: Springer
Year: 1995
Language: English
Pages: 516
Cover......Page 1
Title......Page 2
Copyright......Page 3
Preface to the Second Edition......Page 6
Preface to the First Edition......Page 7
Contents......Page 10
Notation......Page 14
1.1. Binary Operations, Semigroups, and Groups......Page 18
1.2. Examples of Groups......Page 21
1.3. Subgroups and Cosets......Page 25
1.4. Homomorphisms and Quotient Groups......Page 34
1.5. Endomorphisms and Automorphisms......Page 42
1.6. Permutation Groups and Group Actions......Page 48
2.1. Free Groups......Page 61
2.2. Presentations of Groups......Page 67
2.3. Varieties of Groups......Page 73
3.1. Series and Composition Series......Page 80
3.2. Some Simple Groups......Page 88
3.3. Direct Decompositions......Page 97
4.1. Torsion Groups and Divisible Groups......Page 110
4.2. Direct Sums of Cyclic and Quasicyclic Groups......Page 115
4.3. Pure Subgroups and p-Groups......Page 123
4.4. Torsion-Free Groups......Page 131
5.1. Abelian and Central Series......Page 138
5.2. Nilpotent Groups......Page 146
5.3. Groups of Prime-Power Order......Page 156
5.4. Soluble Groups......Page 164
6.1. Further Properties of Free Groups......Page 176
6.2. Free Products of Groups......Page 184
6.3. Subgroups of Free Products......Page 191
6.4. Generalized Free Products......Page 201
7.1. Multiple Transitivity......Page 209
7.2. Primitive Permutation Groups......Page 214
7.3. Classification of Sharply k-Transitive Permutation Groups......Page 220
7.4. The Mathieu Groups......Page 225
8.1. Representations and Modules......Page 230
8.2. Structure of the Group Algebra......Page 240
8.3. Characters......Page 243
8.4. Tensor Products and Representations......Page 252
8.5. Applications to Finite Groups......Page 263
9.1. Hall π-Subgroups......Page 269
9.2. Sylow Systems and System Normalizers......Page 278
9.3. p-Soluble Groups......Page 286
9.4. Supersoluble Groups......Page 291
9.5. Formations......Page 294
10.1. The Transfer Homomorphism......Page 302
10.2. Grün's Theorems......Page 309
10.3. Frobenius's Criterion for p-Nilpotence......Page 312
10.4. Thompson's Criterion for p-Nilpotence......Page 315
10.5. Fixed-Point-Free Automorphisms......Page 322
11.1. Group Extensions and Covering Groups......Page 327
11.2. Homology Groups and Cohomology Groups......Page 343
11.3. The Gruenberg Resolution......Page 350
11.4. Group-Theoretic Interpretations of the (Co)homology Groups......Page 358
12.1. Locally Nilpotent Groups......Page 373
12.2. Some Special Types of Locally Nilpotent Groups......Page 380
12.3. Engel Elements and Engel Groups......Page 386
12.4. Classes of Groups Defined by General Series......Page 393
12.5. Locally Soluble Groups......Page 398
13.1. Joins and Intersections of Subnormal Subgroups......Page 402
13.2. Permutability and Subnormality......Page 410
13.3. The Minimal Condition on Subnormal Subgroups......Page 413
13.4. Groups in Which Normality Is a Transitive Relation......Page 419
13.5. Automorphism Towers and Complete Groups......Page 425
14.1. Finitely Generated Groups and Finitely Presented Groups......Page 433
14.2. Torsion Groups and the Burnside Problems......Page 439
14.3. Locally Finite Groups......Page 446
14.4. 2-Groups with the Maximal or Minimal Condition......Page 454
14.5. Finiteness Properties of Conjugates and Commutators......Page 456
15.1. Soluble Linear Groups......Page 467
15.2. Soluble Groups with Finiteness Conditions on Abelian Subgroups......Page 472
15.3. Finitely Generated Soluble Groups and the Maximal Condition on Normal Subgroups......Page 478
15.4. Finitely Generated Soluble Groups and Residual Finiteness......Page 487
15.5. Finitely Generated Soluble Groups and Their Frattini Subgroups......Page 491
Bibliography......Page 496
Index......Page 508
