New progress on factorized groups and subgroup permutability

Summary

Abstract

The study of products of groups whose factors are linked by certain permutability conditions has been the subject of fruitful investigations by a good number of authors. A particular starting point was the interest in providing criteria for products of supersoluble groups to be supersoluble. We take further previous research on total and mutual permutability by considering significant weaker permutability hypotheses. The aim of this note is to report about new progress on structural properties of factorized groups within the considered topic. As a consequence, we discuss new attainments in the framework of formation theory.

Introduction

In this survey only finite groups are considered.

The study of groups factorized as the product of two subgroups has been the subject of considerable interest in recent years. One of the important questions dealing with this study is how the structure of the factors affects the structure of the whole group and vice versa. A natural approach to this problem is provided by the theory of classes of groups. In this context, the above question can be reformulated as when the belonging of the factors of a factorized group to a class of groups is transferred to the whole group and reciprocally. It is well known that the product of two normal supersoluble subgroups is not supersoluble, in general. Nevertheless, the class of all supersoluble groups U is closed under forming direct and central products. It seems then natural to consider factorized groups in which certain subgroups of the corresponding factors permute, in order to obtain new criteria of supersolubility. A starting point of this research can be located at M. Asaad and A. Shaalan's paper [6]. They considered factorized groups G = AB where A and B are supersoluble subgroups and, in particular, they proved that G is supersoluble under any of the following conditions:

1. (i) Every subgroup of A permutes with every subgroup of B.

2. (ii) A permutes with every subgroup of B, B permutes with every subgroup of A and, moreover, the derived subgroup G’ of G is a nilpotent group.

Products of groups whose factors satisfy condition (i) were called totally permutable products by R. Maier in [33], where he proved that a corresponding result remains valid when the saturated formation U of all supersoluble groups is replaced by any saturated formation containing U.