Article

On the Φ-hypercentral subgroups of finite groups

Published in: Quaestiones Mathematicae
Volume 43 , issue 1 , 2020 , pages: 35–44

DOI: 10.2989/16073606.2018.1533899

Author(s): Hongwei Bao Department of Mathematics, College of Science, People’s Republic of China , Long Miao School of Mathematical Sciences, People’s Republic of China , Huaguo Shi Sichuan Vocational and Technical College, People’s Republic of China , Jia Zhang School of Mathematics and Information, People’s Republic of China

Keywords: 20D10 , 20D20 , 20D10 , 20D20

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Abstract

Assume that is a class of finite groups. A normal subgroup E is Φ- hypercentral in G if E ≤ Z_Φ (G), where Z_Φ (G) denotes the Φ-hypercentre of G. We call a subgroup H is _p-embedded in G, if there exists a p-nilpotent subgroup B of G such that H_p ∈ Syl_p(B) and B is _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(F*(E)) and every noncyclic Sylow p-subgroup P of F*(E), if there is a prime power p^α such that 1 < p^α ≤ | P | and every subgroup H of P with | H| = p^α is _p-embedded in G, then E is Φ-hypercentral in G.

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