FINITE GROUPS WITH WEAKLY S-SEMIPERMUTABLY EMBEDDED SUBGROUPS

Abstract

A subgroup H of G is said to be S-quasinormal in G if H permutes with every Sylow subgroup of G. This concept was introduced by Kegel in 1962 and has been investigated by many authors. A subgroup H is called S-semipermutable in G if H permutes with every Sylow p-subgroup of G for which (p, |H|) = 1. A subgroup H of the group G is said to be c-normal in G if there is a normal subgroup B of G such that HB = G and H ∩ B is normal in G. Next, we unify and generalize the above concepts and give the following concept: A subgroup H of the group G is said to be weakly S-semipermutably embedded in G if there is a subnormal subgroup B of G such that HB = G and H ∩ B is S-semipermutable or S-quasinormally embedded in G. Groups with certain weakly S-semipermutably embedded subgroups of prime power order are studied.