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, $$E\le Z_{{\mathcal {U}}_p}(G)$$ , if there exists a normal subgroup X of G, such that $$F_p^*(E)\le X\le E$$ , and a Sylow p-subgroup P of X satisfies: 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 p-groups 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].