Supersolube subgroups

Abstract

A subgroup H of a group G is termed permutable (or quasi normal) in G if it satisfies the following equivalent conditions:

For any subgroup K of G, HK (the product of subgroups H and K) is a group. For any subgroup K of G, HK= KH, i.e., H and K are permuting subgroups. For every g in G, H permutes with the cyclic subgroup generated by g. Also we say that G=AB is the mutually permutable product of the subgroups A and B if A permutes with every subgroup of B and B permutes with every subgroup of A. We say that the product is totally permutable if every subgroup of A permutes with every subgroup of B. In this paper we prove the following theorem.

Let G=AB be the mutually permutable product of the super soluble subgroups A and B. If CoreG(A∩B)=1, then G is super soluble.