Some new characterizations of solvable PST-groups

A subgroup H of a finite group G is said to be ??-semipermutable in G if it permutes with all the Sylow subgroups Q of G such that (|H|, |Q|) = 1 and (|H|, |Q ^G |) ?? 1. A rather remarkable result of Lukyanenko and Skiba (Rend Semin Mat Univ Padova, 124:231?C246, 2010) is: a finite solvable group G is a PST-group if and only if every subgroup of Fit(G) is ??-semipermutable. A local version of this result is established in this paper. A subgroup H of G is said to be ??-seminormal provided that it is normalized by all Sylow subgroups Q such that (|H|, |Q|) =?1 and (|H|, |Q ^G |) ???1. It is shown that a finite solvable group is a PST-group if and only if every subgroup of Fit(G) is ??-seminormal in G.