A Pushing up Theorem for Groups of Characteristic 2 Type

Abstract

Let G be a finite group with $C_G(O_2(G))\leq O_2(G)$ and S a Sylow 2-subgroup of G. Assume that S is contained in a unique maximal subgroup of G and that no nonidentity characteristic subgroup of S is normal in G. Then it will be shown that G is essentially equal to LMwrT, where L=SL$F_2$(2$F^m$) or $\sum (2^l+1)$, M is the natural GF(2)L-module, and T is a 2-group