On the FΦ-hypercentral subgroups of finite groups

Assume that F is a class of finite groups. A normal subgroup E is FΦ-hypercentral in G if E ≤ ZFΦ(G), where ZFΦ(G) denotes the FΦ-hypercentre of G. We call a subgroup H is Mp-embedded in G, if there exists a p-nilpotent subgroup B of G such that Hp ∈ Sylp(B) and B is Mp-supplemented in G, where Hp is a Sylow p-subgroup of H. In this paper, the main result is that: Let E be a normal subgroup of G. For all p ∈ π (FΦ (E)) and every noncyclic Sylow p-subgroup P of F*(E), if there is a prime power pα such that 1 < pα ≤ l P l and every subgroup H of P with l H l = pα is Mp-embedded in G, then E is UΦ-hypercentral in G.Mathematics Subject Classification (2010): 20D10,20D20.Keywords: FΦ-hypercentral, Mp-embedded, Sylow subgroups, formatio