On the FΦ-hypercentral subgroups of finite groups

Abstract

Assume that F is a class of finite groups. A normal subgroup E is FΦ-hypercentral in G if E ≤ Z_FΦ(G), where Z_FΦ(G) denotes the FΦ-hypercentre of G. We call a subgroup H is M_p-embedded in G, if there exists a p-nilpotent subgroup B of G such that H_p ∈ Syl_p(B) and B is M_p-supplemented in G, where H_p 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 p^α such that 1 < p^α ≤  l P l and every subgroup H of P with l H l = p^α is M_p-embedded in G, then E is UΦ-hypercentral in G.

Mathematics Subject Classification (2010): 20D10,20D20.

Keywords: FΦ-hypercentral, M_p-embedded, Sylow subgroups, formation