Abstract
Subgroups A and B of a finite group are said to be mutually permutable (respectively, M-permutable and-permutable) if A permutes with every subgroup (respectively, every maximal subgroup and every subnormal subgroup) of B and viceversa. If every subgroup of A permutes with every subgroup of B, then the product is said to be totally permutable. These kinds of products have received much attention in the last twenty years. The aim of this paper is to analyse the behaviour of finite pairwise mutually permutable, mutually M-permutable, mutually-permutable and totally permutable products with respect to certain classes of groups including the supersoluble groups, widely supersoluble groups, and also the classes of PST, PT- and T-groups.