Conditional permutability of subgroups and certain classes of groups

Abstract

Two subgroups A and B of a finite group G are said to be tcc-permutable if X permutes with Y g for some g ∈ X, Y , for all X ≤ A and all Y ≤ B. Some aspects about the normal structure of a product of two tcc-permutable subgroups are analyzed. The obtained results allow to study the behaviour of such products in relation with certain classes of groups, namely the class of T-groups and some generalizations.

On a product of two formational tcc-subgroups

“…In the articles [9]- [13] (see also the references from [13]) we can see that the supersolubility of a group can also be obtained for other generalizations of totally permutable product.…”

“…The notation H G means that H is a subgroup of a group G. So, for example, the product G = AB is said to be tcc-permutable [13], if for any X A and Y B there exists an element u ∈ X, Y such that XY u G. The subgroups A and B in this product are called tcc-permutable.…”