p-Supersoluble Hypercenter and s-Semipermutability of Subgroups of a Finite Group

Let G be a finite group and H a subgroup of G. We say that H is s-semipermutable in G if HG(p) = G(p)H for any Sylow p-subgroup G(p) of G with (p, |H|) = 1. In this paper, we consider the s-semipermutability of prime-power order subgroups and prove the following result which generalizes some known results concerning s-semipermutable subgroups. Theorem: Suppose that p is a prime dividing the order of a finite group G and E is a normal subgroup of G. Then, E <= ZU(p) (G), if there exists a normal subgroup X of G, such that F-p* (E) <= X <= E, and a Sylow p-subgroup P of X satisfies: 1. H boolean AND O-p (G*) is s-semipermutable in G for all subgroups H <= P with |H| = d, where d is a power of p with 1 < d < |P|. 2. If p = d = 2 and P is non-abelian, we further suppose H boolean AND O-p (G*) is s-semipermutable in G for H <= P cyclic of order 4. Finally, we also prove a partially generalized version of this theorem, which extends some results in [Y. Berkovich and I. M. Isaacs, p-supersolvability, and actions on pgroups stabilizing certain subgroups, J. Algebra 414 (2014) 82-94.] and [L. Miao, A. Ballester-Bolinches, R. Esteban-Romero, and Y. Li, On the supersoluble hypercentre of a finite group, Monatsh. Math. 184 (2017), no. 4, 641-648].