Math 732 — Spring 2002
Topics in Group Theory

 
Instructor: R. Keith Dennis
Time: MWF 3:45-4:35
Room: Malott 205

The topics to be covered in the course are not yet set in stone. Be the first to cast a vote (or stone) to determine what will be covered. Most likely content is given below.

Option 1: Character Theory of Finite Groups

This would be an introduction to character theory (and hence representation theory) of finite groups. Some easy results on groups proved via character theory are, for example,

  1. Burnside's Theorem: Any finite group of order paqb is solvable.
  2. Frobenius' Theorem: If G is a finite group with subgroup H which is disjoint from all of its conjugates, then the complement of the union of the conjugates with 1 adjoined is a normal subgroup of G.
  3. Burnside: If G is a group of odd order, then the number of conjugacy classes of G is congruent to the group order modulo 16. (Trivial corollaries: A group of order 15 is abelian, a non-abelian group of order 21 has exactly 5 conjugacy classes.)

Suggested references would be
I. M. Isaacs, Character Theory of Finite Groups (Academic Press, 1976; reprint Dover 1994).
B. Huppert, Character Theory of Finite Groups (de Gruyter, 1998).

Option 2: What did Burnside know?

Finite group theory as a subject all its own came into existence with the publication of Burnside's "Theory of Finite Groups" in 1897. (There were earlier books on substitutions and the theory of equations by C. Jordan and E. Netto. A substantial portion of Weber's "Lehrbuch der Algebra" was devoted to the same topic.) Frobenius invented character theory and representation theory in the 1890s and the next 10 years saw a substantial further development of finite group theory. Special cases of the paqb theorem that appeared in the first edition of Burnside's book were replaced by a complete proof via representation theory in the second edition of 1911. The "Notes" section of that edition raised questions and conjectures which were the major impetus for the work in finite group theory during the rest of the century.

During the course we would go through a number of topics as represented by the work of Burnside in his book as well as papers.

Last modified: April 7, 2003