Cornell Math - MATH 739, Fall 2003

MATH 739
Representation Theory and Finite Simple Groups
 (Fall 2003)

Instructor: Gerhard Michler

Meeting Time & Room

Prerequisite: MATH 433 and MATH 631

Other necessary materials from group theory and representation theory will be developed in the course. In particular, an introduction to ordinary and modular representation theory of finite groups will be given.

Aim of the course: Provide foundations from group theory, ordinary and modular representation theory and new algorithms for uniform proofs for existence and uniqueness of the sporadic simple groups. Show that these methods can also be used to prove the uniqueness for simple groups belonging to infinite series.

Topics

  1. This course starts with the famous Brauer-Fowler Theorem: Let H be a finite group of even order. Then there are only finitely many simple groups G having an element z not equal to 1 of order 2 in the center

  2. Z(S) = {x in S| xs = sx} for some Sylow 2-subgroup S of G.

    If there are several target groups G, then they are called satellites. If no proper satellites exist, then the simple group G is uniquely determined.

  3. Then a recent (theoretical and in concrete situations practical) algorithm is presented for constructing the possible target groups G from a given set of generators and relations of H.
  4. The satellites PSL2(5) and the sporadic simple Held He group of the Mathieu group M24 will be constructed by this algorithm, after the definitions of the Mathieu groups have been given.
     

    Thus the basic tools for calculations and proofs of the character tables of the large sporadic groups are provided.

  5. In order to compute the character table and the conjugacy classes of a target group G a survey is given about algorithms for calculating in permutation groups. A recent general character formula of M. Weller and G. Michler will be proved.

    Therefore a new general uniqueness criterion will be proved. For its applications the following classical results of R. Brauer in ordinary and modular representation theory will be presented:

    (a) His characterization of characters

    (b) His main theorems in modular representations

    (c) His and Thompson's group order formulas

  6. For the classification of the finite simple groups the uniqueness question is of great importance, see M. Suzuki, Group Theory II, Springer Verlag, Heidelberg (1986), p. 421.
  7. Uniform existence and uniqueness proofs for sporadic simple groups
     
  8. On the uniqueness of the alternating groups An, n greater than or equal to 5.
     
  9. Open research problems.

Lecture notes (with complete proofs) can be provided to the participants of the course.