MATH 7350: Topics in Group Theory (Spring 2010)

Instructor: R. Keith Dennis

Prerequisites: Basics of algebra, in particular group theory (e.g., MATH 4340, MATH 6310)

The course might well be titled “a second course in group theory” as basic results about groups (groups acting on sets, Sylow theorems, etc.) will be assumed. Any other necessary background from elementary group theory (wreath products,...) will be summarized or developed in class as needed.

This course will be accessible to beginning graduate students and advanced undergraduates — should be of interest to those interested in algebra, representation theory, K-theory, algebraic topology, among others.

Likely topics to be covered are 1, 2, 3, 4:

  1. Basic stuff: Universal properties. Exact sequences of groups, split extensions, direct products, central products. semi-direct products, complete groups, wreath products, cocycles, second cohomology group, ...
  2. Schur-Zassenhaus Theorem.
  3. Wedderburn-Krull-Remak-Schmidt Theorem.
  4. Hall's theorems on solvable groups.
  5. Permutation groups
  6. Mathieu groups
  7. Transfer and fusion, Alperin's theorem
  8. Thompson subgroup, Thompson complement theorem
  9. Group characters
  10. Frobenius groups
  11. The Moebius function of finite groups.
  12. Varieties of groups.

Other topics, e.g., 5-12 (or others) will be covered depending on the interest of those attending. Please ask!

Text: None. Several references might be useful: e.g.,

  • J. D. Dixon & B. Mortimer, Permutation groups, GTM 163, Springer, 1996.
  • L. Grove, Groups and characters, Wiley, 1997.
  • L. Grove, Classical Groups and Geometric Algebra, AMS, 2001.
  • M. Hall, The theory of groups, MacMillan.
  • H. Kurzweil and B. Stellmacher, The theory of finite groups, Springer.
  • H. Neumann, Varieties of groups, Springer.
  • J. Rotman, An introduction to the theory of groups, Springer.
  • M. Suzuki, Group theory I, II, Springer.
  • B. A. F. Wehrfritz, Finite groups: A second course on group theory, World Scientific, 1999.