Transitivity of Sylow permutability, the converse of Lagrange's theorem, 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 Y -groups (groups satisfying the converse of Lagrange's theorem) and SC-groups (groups whose chief factors are simple) are SC -groups. Next, we show that a product of pairwise mutually permutable Y -groups is supersoluble. Finally, we give a local version of the result stating that if 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 -group