The aim of this book is to provide a concise treatment of some topics from group theory and representation theory for a one term course. It focuses on the non-commutative side of the field emphasizing the general linear group as the most important group and example. The book will enable graduate students from every mathematical field, as well as strong undergraduates with an interest in algebra, to solidify their knowledge of group theory. The reader should have a familiarity with groups, rings, and fields, along with a solid knowledge of linear algebra. Close to 200 exercises of varying difficulty serve both to reinforce the main concept of the text and to expose the reader to additional topics
CONTENT
1. Rudiments of Group Theory -- 1. Review -- 2. Automorphisms -- 3. Group Actions -- 2. The General Linear Group -- 4. Basic Structure -- 5. Parabolic Subgroups -- 6. The Special Linear Group -- 3. Local Structure -- 7. Sylowโ{128}{153}s Theorem -- 8. Finite p-groups -- 9. The Schur-Zassenhaus Theorem -- 4. Normal Structure -- 10. Composition Series -- 11. Solvable Groups -- 5. Semisimple Algebras -- 12. Modules and Representations -- 13. Wedderburn Theory -- 6. Group Representations -- 14. Characters -- 15. The Character Table -- 16. Induction -- Appendix: Algebraic Integers and Characters -- List of Notation
