A characterization of finite groups containing a strongly closed 2-subgroup

Abstract

Throughout this paper G is a finite group and S is a subgroup of G. When S is contained in the subgroup H of G, we say S is strongly closed in H with respect to G if whenever s ∈ S and g ∈ G are such that s ∈ H, then s ∈ S. In other words, every G-conjugacy class of elements of S intersected with H is contained in S. In the case where S is a p-group for some prime p we say that S is strongly closed if it is strongly closed in some Sylow p-subgroup containing it. One easily sees that S is strongly closed if and only if it is strongly closed in NG(S) with respect to G, so strong closure is independent of the choice of Sylow p-subgroup containing S.