Criteria for Permutability to Be Transitive in Finite Groups

A group G is a PT-group if, for subgroups H and K with H permutable in K and K permutable in G, it is always the case that H is permutable in G. It is shown that a finite group is a soluble PT-group if and only if each subgroup of a Sylow subgroup is permutable in the Sylow normalizer.