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 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 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
