2-Fusion in Finite Groups

Abstract

Suppose A c T c G are groups such that whenever a E A, g e G, and ag E T, then as e A. In this situation, we say that A is strongly closed in T with respect to G. We are concerned here with the case where G is finite, T is a Sylow 2-subgroup of G, and A is Abelian. In this case, we call G an S(A)-group. Our objective is to determine the composition factors of AG, the normal closure of A in G, when G is an S(A)-group. For this purpose, we make the following definitions: Suppose L is a perfect group such that L/Z(L) is simple. (Such a group is called quasi-simple.) We shall say that L is of type I provided L/Z(L) is isomorphic to one of the following: (a) L2(2), n > 3. (b) Sz(22n+l), n~ > 1. (c) U3 (2 ), n > 2. We shall say that L is of type II provided Z(L) has odd order and L = L/Z(L) satisfies one of the following conditions: (d) L L2(q), q 3, 5 (mod 8). (e) L is simple and contains an involution t such that (1) I L: CL(t) I is odd, (2) CL(t) = x K, (3) K contains a normal subgroup E of odd index such that CK(E) = 1 and E L2(q)g q _ 3, 5 (mod 8).