Groups in which Sylow subgroups and subnormal subgroups permute

Abstract

We consider certain properties of finite groups in which the subnormal subgroups permute with all the Sylow subgroups. Such groups are called PST-groups. If $G$ is such a group and $ H_1 / K_1 $ and $ H_2 / K_2 $ are isomorphic abelian chief factors of $G$ such that $ H_1 H_2 \subseteq G' $, then they are operator isomorphic. Moreover, if all the abelian isomorphic chief factors of a PST-group $G$ are operator isomorphic, then all the subnormal subgroups are hypercentrally embedded in $G$.