Weakly $S$-semipermutable subgroups and $p$-nilpotency of groups

A subgroup H of a finite group G is said to be S-semipermutable in G if HGp = GpH for every Sylow subgroup Gp of G with (|H|, p) = 1. A subgroup H of G is said to be Weakly S-semipermutable in G if there exists a normal subgroup T of G such that HT is S-permutable and H ∩ T is S-semipermutable in G. In this paper we prove that for a finite group G, if some cyclic subgroups or maximal subgroups of G are Weakly S-semipermutable in G, then G is p-nilpotent. Mathematics Subject Classification (2010). Primary: 20D15; Secondary: 20D20, 20F19, 20D10.