Conditional permutability of subgroups and certain classes of groups

Two subgroups A and B of a finite group G are said to be tcc-permutable if X permutes with Yg 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.