On weakly S-semipermutable subgroups and weakly SS-permutable subgroups of finite groups

A subgroup H of a finite group G is said to be S-permutable in G, if H permutes with all Sylow subgroups of G. If H permutes with every Sylow p-subgroup of G such that (|H|; p) = 1, then H is called an S-semipermutable subgroup of G. In this note, we say that H is a weakly S-semipermutable subgroup of G, if there exists a normal subgroup T of G such that HT is Spermutable and H ∩ T is S-semipermutable in G. We assume that some p-subgroups of G are weakly S-semipermutable in G and we verify the influence of these subgroups on the structure of G.