Finite groups with pro-normal subgroups

Abstract

The purpose of this paper is to study a class of finite groups whose subgroups of prime power order are all pro-normal. Following P. Hall, we say that a subgroup H of a group G is pro-normal in G if and only if, for all x in G, H and Hz are conjugate in (H, HX), the subgroup generated by H and HX. We shall determine the structure of all finite groups whose subgroups of prime power order are pro-normal. It turns out that these groups are in fact soluble t-groups. A t-group is a group G whose subnormal subgroups are all normal in G. The structure of finite soluble t-groups has been determined by Gaschuitz [1]. Are soluble t-groups precisely those, groups whose subgroups of prime power order are all pro-normal? The answer is yes. Thus our study furnishes another characterization of soluble t-groups. Except for the definitions given above, our terminology and notation are standard (see, for example, M. Hall [2 ]). We first prove two general facts about pro-normal subgroups.