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

Abstract

There is a long running project due to U. Meierfrankenfeld and the author to investigate the so called small modules for the finite simple groups. These modules show up in the amalgam method which recently became important for the revision of parts of the classification of the finite simple groups. A small module either is a quadratic module or a module on which an elementary abelian group acts such that the codimension of the centralizer is small compared with its order. In this paper we determine all irreducible modules $V$ over $GF(2)$ for the finite simple groups $G$ such that $|V : C_V(A)| \le |A|^2$ for some nontrivial elementary abelian subgroup $A$ of $G$ where in addition we have $[V, A, A] = 1$.