# Small representations of finite simple groups

## Description of the research project

Let G be a finite group such that F^*(G) is quasisimple. We are interested in classifying so called small modules for the group G. These modules play in the revision of the classification of the finite simple groups and in the amalgam method in general

Let G be a finite group such that F^*(G) is quasisimple. We are interested in classifying so called small modules for the group G. These modules play a prominent role in the revision of the classification of the finite simple groups and in the amalgam method in general. The concept of a small module is not well defined. We basically have four classes of modules. Let $V$ be a faithful GF(p)-module for G, then

1) quadratic modules V : i.e. there is some nontrivial elementary abelian p-subgroup A of G such that [V,A,A] = 0, in case of p = 2 we ask |A| > 2.

2) F - modules V : i.e. there is some nontrivial elementary abelian p-subgroup A of G such that

|V : C_V(A)| \leq |A|.

3) F+1 -modules V : .i.e. there is some nontrivial elementary abelian p-subgroup A of G such that

|V : C_V(A)| \leq 2|A|

4) 2F+1 - modules V : there is some nontrivial elementary abelian p-subgroup A of G such that

|V : C_V(A)| \leq 2|A|^2

5) 2F - modules with quadratic offender : there is some nontrivial elementary abelian p-subgroup A of G such that

|V : C_V(A)| \leq |A|^2 and $[V,A,A] = 1$

There are many results about the groups in 1) (missing are Chevalley groups in even characteristic, but there are good results for those close to the general case). The groups in 2) are done. There are many results in 3) and 4). In particular if G is simple amd p = 2.

## This is joint work partly done together with

U. Meierfrankenfeld (East-Lansing, MI, USA)

The following preprint related to the project may be viewed by clicking below:

Latex File

• F-modules for finite simple groups (Joint with Ulrich Meierfrankenfeld)

Latex File

• 2F-modules with quadratic offender for the finite simple groups

Latex File

• F+1 and 2F+1-modules for finite simple groups (Joint with Ulrich Meierfrankenfeld)

Latex File in preparation