A Course in Group Theory [Paperback]

The classification of the finite simple groups is one of the major intellectual achievements of this century, but it remains almost completely unknown outside of the mathematics community. This introduction to group theory is also an attempt to make this important work better known. Emphasizing classification themes throughout, the book gives a clear and comprehensive introduction to groups and covers all topics likely to be encountered in an undergraduate course. Introductory chapters explain the concepts of group, subgroup and normal subgroup, and quotient group. The homomorphism and isomorphism theorems are explained, along with an introduction to G-sets. Subsequent chapters deal with finite abelian groups, the Jordan-Holder theorem, soluble groups,p-groups, and group extensions. The numerous worked examples and exercises in this excellent and self-contained introduction will also encourage undergraduates (and first year graduates) to further study.

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-H??lder 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
Indexl³;