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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 BaoDepartment of Mathematics, College of Science, People’s Republic of China, Long MiaoSchool of Mathematical Sciences, People’s Republic of China, Huaguo ShiSichuan Vocational and Technical College, People’s Republic of China, Jia ZhangSchool of Mathematics and Information, People’s Republic of China

Abstract

Assume that is a class of finite groups. A normal subgroup E is Φ- hypercentral in G if EZΦ (G), where ZΦ (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 HpSylp(B) and B is p-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(F*(E)) and every noncyclic Sylow p-subgroup P of F*(E), if there is a prime power pα such that 1 < pα ≤ | P | and every subgroup H of P with | H| = pα is p-embedded in G, then E is Φ-hypercentral in G.

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