… In this paper a local version of Agrawal's theorem about the structure of finite groups in
which Sylow permutability is transitive is given. The result is used to obtain new
characterisations of this class of finite groups. … We prove in the following that the
permutability with the Carter subgroups can be removed. …

… This result is useful to get information about the classes of groups in which every
subnormal subgroup is permutable and Sylow permutable.  2002 Elsevier Science (USA)
Key Words: … For instance, it is proved in [5, Theorem H] that a soluble group G is a PST-group
if and only if every subnormal subgroup permutes with every Carter subgroup of G and the
subnormal subgroups are hypercentrally embedded in G. As an application of Theorem 5,
we prove in [3, Corollary 2] that the permutability with the Carter subgroups can be removed. …