# Group Theory A5

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Though these notes require no previous knowledge of group theory it is more suitable for a third year student than a first year one because it goes into the subject quite deeply. The opening chapters attempt to motivate group theory with many illustrative examples such as shuffling of cards, bell ringing and permutation puzzles.Fairly early on the student is introduced to group presentations and the Todd-Coxeter algorithm. Although this algorithm has been around for about 80 years it is not normally taught. However as it underlies the modern approach to group theory it is well worth students knowing it. A new framework for the algorithm makes it accessible to third-year students.

Another novelty, for a first course in group theory, is the inclusion of a couple of chapters on representation theory. Although a full stand-alone treatment would be quite deep, by assuming the Fundamental Theorem of Characters (the irreducible characters forming an orthonormal basis) everything else that is needed can be easily proved. Students seem to enjoy the challenge of constructing character tables from a group presentation and the theory of characters reinforces the concepts and methods of linear algebra.

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- Introduction and Contents
- CHAP01 Introduction to Groups
- CHAP02 Permutations
- CHAP03 Examples of Groups
- CHAP04 First Steps in Group Theory
- CHAP05 The Todd-Coxeter Algorithm
- CHAP06 A Second Round of Theory
- CHAP07 Representations of Finite Groups
- CHAP08 Inducing Characters
- CHAP09 Free Groups
- CHAP10 Finitely-Generated Abelian Groups
- CHAP11 Soluble Groups
- CHAP12 Infinite Abelian Groups
- CHAP13 Sylow Subgroups
- CHAP14 Nilpotent Groups
- CHAP15 Automorphisms
- CHAP16 Class Equations
- CHAP17 Hall Subgroups
- APPENDIX A Groups of Small Order
- APPENDIX B Character Table Summary
- APPENDIX C Biographies of Some Group Theorists
- APPENDIX D Projects