A note on the embedding properties of p-subgroups in finite groups

Abstract

In this note, we prove that a finite group G is p-supersolvable if and only if there exists a power d of p with p2 ≤ d d if there exists a power d of p with 1 ≤ d < |P| such that H ∩Op(G∗ p2) is normal in Op(G) for all non-meta-cyclic normal subgroups H of P with |H|= d.