Finite groups with abelian sylow 2-subgroups of order 8

This paper provides a result necessary in the classification of simple finite groups with abelian S2-subgroups. In classifying these groups in another paper, we shall determine the structure of the centralizers of the involutions in such groups (cf. [24]). In particular, for those groups which have S2-subgroups of order 8, we shall verify the hypothesis of the principal theorem of this paper. Then from this theorem follows the conditions by which the recent papers of JANKO and THOMPSON [16], JANKO [15], and WARD [25] show that the simple group in question is either the one recently discovered by JANKO (cf. [15]) or has the character table given by Ward of the simple group of automorphisms of the simple Lie algebra of type (7 2 over a finite field of characteristic 3 discovered by REE [18]. The latter type of group is defined in Section 1 as an R-group. Theorem. Let (5 be a group with abelian S2-subgroups of order 8. Let = (J) be the subgroup generated by an involution J. Suppose that the centralizer .~=C~(J) of J in (5 contains a normal subgroup which is isomorphic to 1 PSL(2, q). Then one of the following holds.