Introduction

All groups considered in this note are finite. If H is a subgroup of a group G, we write and Recall that Subgroups which permute with all Sylow subgroups of the group, or S-permutable subgroups, were introduced by Kegel in his seminal paper [K 62]. P. Schmid, in [Sch 98], presented an extensive and elegant study of these subgroups. In that paper it is proved that for a core-free S-permutable subgroup T of a group G which also permutes with the normalizer of a Sylow subgroup N, then T ≤ N ([Sch 98]; Prop. D). Since the hypercenter Z∞(G) of the group G, i.e. the last member of the ascending central series of G, is the intersection of the normalizers of all Sylow subgroups of G, we have that for a subgroup T of G the following are equivalent:

1. (i) T is an S-permutable subgroup which permutes with the normalizers of all Sylow subgroups of G, and

2. (ii) TG/TG ≤ Z∞(G/TG), for TG = ∩g∈GTg, the core of T in G.

Such a subgroup is said to be a hypercentrally embedded subgroup. We focus our attention on these special S-permutable subgroups. Hypercentrally embedded subgroups enjoy very good factorization properties. In fact, previously, Carocca and Maier, in [CM 98], had characterized hypercentrally embedded subgroups as those subgroups which permute with all pronormal subgroups. In this note we present some factorizations of hypercentrally embedded subgroups with some special types of subgroups which, in general, are not pronormal.