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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 ). The purpose of this paper is to prove a further generalization of these results: i.e. to prove the following theorem.Main Theorem. Let G be a finite group of even order. If the centralizer of any involution in G is always abelian then we have one of the following three possibilities: (1) 2-Sylow subgroups of G are cyclic, (2) a 2-Sylow subgroup of G is a normal subgroup, or (3) G is a direct product of two groups L and A where L is one of the linear groups Lq with q = 2" and A is an abelian group of odd order.The proof of this result is rather complicated.The first section is devoted to a reduction process. Assuming our main theorem to be false we study the structure of the group with the smallest order for which the assertion of the main theorem is not true. The second section is concerned with a particular type of finite groups with structure similar to the one obtained at the end of the first section. In view of further applications we take slightly more general assumptions. Our main purpose is to obtain a formula for the order of such a group which is applied in the next section to prove the nonexistence of the group considered in §1. The method used here is similar to the one in [l] or [6]. An application of our main theorem will be given in the final section.