Notation and terminology

A group G is said to be a T-group if every subnormal subgroup of G is normal in G, that is, if normality is a transitive relation. These groups have been widely studied (see [10], [11], or [14]).

A subgroup H of a group G is said to be permutable (or quasinormal) in G if HK = KH for all subgroups K of G. Permutability can be considered thus as a weak form of normality. The study of groups G in which permutability is transitive, that is, H permutable in K and K permutable in G always imply that H is permutable in G, has been a successful field of research in recent years. Such groups are called PT-groups. According to a theorem of Kegel [12, Satz 1], every permutable subgroup of G is subnormal in G. Consequently, PT-groups are exactly those groups in which subnormality and permutability coincide; that is, those groups in which every subnormal subgroup permutes with every other subgroup. Therefore, every T-group is clearly a PT-group.

One could wonder what would happen if we did not require that every subnormal subgroup of a group G permutes with any other subgroup of G, but only with a certain family of its subgroups. In this direction, those groups in which every subnormal subgroup of G permutes with every Sylow p-subgroup of G for each prime p have sometimes been called T*-groups (see [3]) or also (π – q)-groups (see [1]).