On FΦ*-hypercentral subgroups of finite groups

[摘要] Let G be a finite group. We write R(G) to denote the largest soluble normal subgroup of G and put Phi*(G) = Phi(R(G)). We say that a chief factor H/K of G is non-Frattini (non-solubly-Frattini) if H/K not less than or equal to Phi(G/K) (if H/K not less than or equal to Phi*(G/K), respectively). A chief factor H/K of G is called T-central in G provided (H/K) (sic) (G/C-G(H/K)) is an element of F. A normal subgroup N of G is said to be F Phi-hypercentral (F Phi*-hypercentral) in G if either N = 1 or N not equal 1 and there exists a chief series 1 = N-0 < N-1 < .... < N-t = N (*) of G below N such that every non-Frattini (non-solubly-Frattini, respectively) factor N-i/Ni-1 of Series (*) is F-central in G. In this paper we analyze some properties and applications of F Phi-hypercentral and F Phi*-hypercentral subgroups. (C) 2012 Elsevier Inc. All rights reserved.

[发布日期] 2012-12-15

[关键词] Finite group;F Phi-hypercentral subgroup;F Phi*-hypercentral subgroup;Saturated formation;Solubly saturated formation;Quasinilpotent group