On characterizations of linear groups. I

Abstract

Let Lq be the totality of nonsingular 2X2 matrices with determinant 1 over a finite field of q elements. This paper is concerned with a grouptheoretical characterization of Lq in the case that q is a power of 2. We shall call an element of order 2 an involution. It is easy to see that Lq (q = 2n) contains an involution t such that the centralizer of r in Lq is an abelian 2-group. In his thesis [5] K. A. Fowler proved that this property (together with nonsolvability) actually characterizes Lq. A few years later working independently G. E. Wall and the author obtained a generalization of Fowler's theorem characterizing Lq under weaker assumptions (cf. [3]). The purpose of this paper is to prove a further generalization of these results: i.e. to prove the following theorem.