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

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 < |P| such that H ∩ Op(G∗ p) is normal in Op(G) for all non-cyclic normal subgroups H of P with |H|= d, where P is a Sylow p-subgroup of G. Moreover, we also prove that either lp(G) ≤ 1 and rp(G) ≤ 2 or else |P ∩Op(G)| > 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.

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