This book is an excellent and self-contained introduction to the theory of groups, covering all topics likely to be encountered in undergraduate courses. It aims to stimulate and encourage undergraduates to find out more about their subject. The book takes as its theme the various fundamental classification theorems in finite group theory, and the text is further explained in numerous examples and exercises, and summaries at the end of each chapter.

1. Definitions and examples 2. Maps and relations on sets 3. Elementary consequences of the definitions 4. Subgroups 5. Cosets and Lagrange's Theorem 6. Error-correcting codes 7. Normal subgroups and quotient groups 8. The Homomorphism Theorem 9. Permutations 10. The Orbit-Stabilizer Theorem 11. The Sylow Theorems 12. Applications of Sylow Theorems 13. Direct products 14. The classification of finite abelian groups 15. The Jordan-Holder Theorem 16. Composition factors and chief factors 17. Soluble groups 18. Examples of soluble groups 19. Semi-direct products and wreath products 20. Extensions 21. Central and cyclic extensions 22. Groups with at most 31 elements 23. The projective special linear groups 24. The Mathieu groups 25. The classification of finite simple groups Appendix A Prerequisites from Number Theory and Linear Algebra Appendix B Groups of order < 32 Appendix C Solutions to Exercises Bibliography Index

John F. HumphreysDepartment of Pure Mathematics, University of Liverpool

`The arguments are clear and full proofs are given. ... The whole text is actually built up around the idea of classification theorems. The inherent limitations of such an approach put aside, such glimpses of a distant horizon can do a lot towards stimulating the students to find more about the subject for themselves.' M Deaconescu, Zentralblatt fur Mathematik, Band 843/96. |d 14/10/1997