Properties of Quasinormal Groups (PQG)

Abstract

The A subgroup of a group  is termed permutable (or quasinormal) in  if it satisfies the following equivalent conditions:

1. For any subgroup of  ,  (the product of subgroups  and  ) is a group


2. For any subgroup of ,  , i.e.,  and  are permuting subgroups.


3. For every ,  permutes with the cyclic subgroup generated by  . In symbols, for every  and  , there exists  and an integer  such that  .

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.