Index two simple groups

Abstract

A group, G, is an index two simple group if G is a finite simple group whose order is of the form pm where p is a prime, ( p, m) = 1 and the index of a Sylow p-subgroup in its normalizer is 2. In this paper the theory of index two simple groups is developed and applied. In particular index two simple groups are shown to have exactly one conjugacy class of involutions and a sharp bound for the order of such a group is found as a function of the degree, n, of a nonidentity ordinary irreducible character in the principal p-block. It is shown that the only index two simple groups with n ≤ 25 are PSL(2, 5), PSL(2, 7), PSL(2, 8), PSL(2, 9), PSL(2, 11), PSL(2, 13), PSL(2, 16), PSL(2, 23) and PSL(2, 25).