ON ASCHBACHERS LOCAL C(G,T) THEOREM

Aschbacher's local C(G;T) theorem asserts that if G is a finite group with F(G) = O-2(G), and T is an element of SYl(2)(G), then G = C(G;T) K(G), where C(G;T) = [N-G(T-0) \ 1 not equal T-0 char T] and K(G) is t he product of all near components of G of type L(2)(2(n)) or A(2n+1). Near components are also known as <chi-blocks or Aschbacher blocks. In this paper we give a proof of Aschbacher's theorem in the case that G is a K-group, i.e., in the case that every simple section of G is iso morphic to one of the known simple groups. Our proof relies on a resul t of Meierfrankenfeld and Stroth [MS] on quadratic four-groups and on the Baumann-Glauberman-Niles theorem, for which Stellmacher [St2] has given an amalgam-theoretic proof. Apart from those results, our proof is essentially self-contained.