This book quickly introduces beginners to general group theory and then focuses on three main themes : finite group theory, including sporadic groups combinatorial and geometric group theory, including the Bass-Serre theory of groups acting on trees the theory of train tracks by Bestvina and Handel for automorphisms of free groups With its many examples, exercises, and full solutions to selected exercises, this text provides a gentle introduction that is ideal for self-study and an excellent preparation for applications. A distinguished feature of the presentation is that algebraic and geometric techniques are balanced. The beautiful theory of train tracks is illustrated by two nontrivial examples. Presupposing only a basic knowledge of algebra, the book is addressed to anyone interested in group theory: from advanced undergraduate and graduate students to specialists.
Author(s): Oleg Bogopolski
Series: EMS Textbooks in Mathematics
Edition: 1
Publisher: European Mathematical Society
Year: 2008
Language: English
Pages: 189
Cover......Page 1
Preface......Page 6
Preface to the Russian Edition......Page 7
Contents......Page 10
Main definitions......Page 12
Lagrange's theorem. Normal subgroups and factor groups......Page 15
Homomorphism theorems......Page 17
Cayley's theorem......Page 18
Double cosets......Page 20
Actions of groups on sets......Page 21
Normalizers and centralizers. The centers of finite p-groups......Page 23
Sylow's theorem......Page 24
Direct products of groups......Page 26
Finite simple groups......Page 27
The simplicity of the alternating group A_n for n > 5......Page 29
A_5 as the rotation group of an icosahedron......Page 30
A_5 as the first noncyclic simple group......Page 31
A_5 as a projective special linear group......Page 33
A theorem of Jordan and Dickson......Page 34
Mathieu's group M_22......Page 36
The Mathieu groups, Steiner systems and coding theory......Page 43
Extension theory......Page 46
Schur's theorem......Page 48
The Higman–Sims group......Page 50
Graphs and Cayley's graphs......Page 56
Automorphisms of trees......Page 61
Free groups......Page 63
The fundamental group of a graph......Page 67
Presentation of groups by generators and relations......Page 69
Tietze transformations......Page 71
A presentation of the group S_n......Page 74
Trees and free groups......Page 75
The rewriting process of Reidemeister–Schreier......Page 80
Free products......Page 82
Amalgamated free products......Page 83
Trees and amalgamated free products......Page 85
Action of the group SL_2(Z) on the hyperbolic plane......Page 87
HNN extensions......Page 92
Graphs of groups and their fundamental groups......Page 95
The relationship between amalgamated products and HNN extensions......Page 98
The structure of a group acting on a tree......Page 99
Kurosh's theorem......Page 102
Coverings of graphs......Page 104
S-graphs and subgroups of free groups......Page 107
Foldings......Page 109
The intersection of two subgroups of a free group......Page 112
Complexes......Page 115
Coverings of complexes......Page 117
Surfaces......Page 120
Grushko's Theorem......Page 126
Hopfian groups and residually finite groups......Page 128
3 Automorphisms of free groups and train tracks......Page 132
Nielsen's method and generators of Aut(F_n)......Page 134
Maps of graphs. Tightening, collapsing and expanding......Page 137
Homotopy equivalences......Page 139
Topological representatives......Page 140
The transition matrix. Irreducible maps and automorphisms......Page 141
Transformations of maps......Page 143
The metric induced on a graph by an irreducible map......Page 148
Proof of the main theorem......Page 149
Examples of the construction of train tracks......Page 152
Two applications of train tracks......Page 162
Appendix. The Perron–Frobenius Theorem......Page 164
Solutions to selected exercises......Page 168
Bibliography......Page 180
Index......Page 184
Back Cover......Page 189
