A note on finite 𝒫𝒮𝒯-groups

A finite group G is said to be a 𝒫𝒮𝒯-group if, for subgroups H and K of G with H Sylow-permutable in K and K Sylow-permutable in G , it is always the case that H is Sylowpermutable in G . A group G is a 𝒯*-group if, for subgroups H and K of G with H normal in K and K normal in G , it is always the case that H is Sylow-permutable in G . In this paper, we show that the classes of finite 𝒫𝒮𝒯-groups and finite 𝒯*-groups coincide. A new characterization of soluble 𝒫𝒮𝒯-groups is also presented.