Some commutation patterns between involutions of a finite group. I

Abstract

Let G be a finite group generated by its involutions. A subset D is said to be 2-powerful in G provided D contains some involutions, and every involution in G commutes with some involution from D; for instance, let D meet nontrivially the center of every Sa-subgroup of G. In this paper, we consider the case where D = {a, b 1 un = b2 = (c~b)~ = 1) is a dihedral subgroup of G of order 2n, n odd. This presentation for D will be maintained throughout the paper. Our first theorem describes an aspect of the local structure of G. THEOREM 3. Let L be an involution generated subgroup of G such that: D CL C G, L normalizes and is represented faithfully on V, where V is an elementary abelian 2-subgroup of G. Furthermore, suppose that C,(ai) is trivial for all i +k 0 (mod n). Then, there exists a positive integer m such that (i) n = 2” + 1, / V 1 = 22nt, (ii) there exists a jield F contained in End(V) such that F z GF(2”), and L C SL( V, F), and (iii) V _C O,(G). For the special values n = 3, 5 we obtain the following results.