Some Classes of Finite Supersoluble Groups

In this survey we study the relation between the class of groups in which Sylow permutability is a transitive relation (the PST-groups) and the class of groups in which every subgroup possesses supergroups of all possible indices, the so-called $\mathcal{Y}$-groups. The parellelism between these classes in the soluble universe and the interest of the local study of PST-groups motivates a local study of  $\mathcal{Y}$-groups.

A group $G$ factorised as a product of two subgroups $A$ and $B$ is said to be a mutually permutable product whenever $A$  permutes with every subgroup of  $B$ and $B$ permutes with every subgroup of $A$ . We present some results concerning mutually permutable products of groups in the orbit of the above classes.