Some classes of finite groups and mutually permutable products

This paper is devoted to the study of mutually permutable products of finite groups. A factorised group G=AB is said to be a mutually permutable product of its factors A and B when each factor permutes with every subgroup of the other factor. We prove that mutually permutable products of -groups (groups satisfying a converse of Lagrangeʹs theorem) and SC-groups (groups whose chief factors are simple) are SC-groups, by means of a local version. Next we show that the product of pairwise mutually permutable -groups is supersoluble. Finally, we give a local version of the result stating that when a mutually permutable product of two groups is a PST-group (that is, a group in which every subnormal subgroup permutes with all Sylow subgroups), then both factors are PST-groups.