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 $\mathcal Y$-groups (groups satisfying the converse of Lagrange's theorem) and $\mathrm {SC}$-groups (groups whose chief factors are simple) are $\mathrm{SC}$-groups. Next, we show that a product of pairwise mutually permutable $\mathcal Y$-groups is supersoluble. Finally, we give a local version of the result stating that if a mutually permutable product of two groups is a $\mathrm{PST}$-group (that is, a group in which every subnormal subgroup permutes with all Sylow subgroups), then both factors are $\mathrm{PST}$-groups.