Finite groups with many S-semipermutable p-subgroups

A subgroup H of a finite group G is said to be S-semipermutable in G if H permutes with every Sylow subgroup of G of order coprime to it. Huppert obtained the structure and properties of finite groups with all Sylow subgroups S-semipermutable, Wang, Li and Wang gave the structure and properties of finite groups with all subgroups S-semipermutable. The main purpose of this paper is to give local versions of these results. We consider the structure and properties of groups G with all Sylow p-subgroups S-semipermutable and of groups G with all p-subgroups S-semipermutable, where p is a fixed prime dividing the order of G.