Journal of Material Sciences and Engineering Technology

Products of Quasinormal Groups (QSG)

Behnam Razzaghmanesshi

ABSTRACT
A subgroup H of a group G is termed quasinormal in G if it satisfies the following equivalent conditions:

1. For any subgroup K of G, HK (the product of subgroups H and K) is a group
2. For any subgroup K of G, HK=KH, i.e., H and K are permuting subgroups.
3. For every  gG, H permutes with the cyclic subgroup generated by g. In symbols, for every hH and gG, there exists h1H and an integer n such that hg = gnh1.
4. 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 Core G (A∩B)=1, then G is super soluble.