Research ArticleNo Access

On $Π$ $Π$ -property of subgroups of a finite group

Xinwei Wu

School of Mathematical Science, Soochow University, Suzhou, Jiangsu 215006, P. R. China

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and

Xianhua Li

https://orcid.org/0000-0003-0571-5278

School of Mathematical Science, Soochow University, Suzhou, Jiangsu 215006, P. R. China

E-mail Address: xhli@suda.edu.cn

Corresponding author.

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https://doi.org/10.1142/S0219498822501560Cited by:0 (Source: Crossref)

Abstract

Let $G$ $G$ be a finite group, $p$ $p$ a prime, $P$ $P$ a Sylow $p$ $p$ -subgroup of $G$ $G$ and $d$ $d$ a power of $p$ $p$ such that $1 < d < | P |$ $1 < d < | P |$ . Let $G_{p}^{*}$ $G_{p}^{*}$ denote the unique smallest normal subgroup of $G$ $G$ for which the corresponding factor group is abelian of exponent dividing $p - 1$ $p - 1$ . Let $𝔉_{1}$ , $𝔉_{2}$ , $𝔉_{3}$ be classes of all $p$ -groups, $p$ -nilpotent groups and $p$ -supersolvable groups, respectively, $G^{𝔉}$ be the $𝔉$ -residual of $G$ . Let $X \in {{(G_{p}^{*})}^{𝔉_{1}}, G^{𝔉_{2}}, G^{𝔉_{3}}}$ . A subgroup $H$ of a finite group $G$ is said to have $Π$ -property in $G$ , if for any $G$ -chief factor $L / K$ , $| G / K : N_{G / K} ((H \cap L) K / K) |$ is a $π ((H \cap L) K / K)$ -number. A normal subgroup $E$ of $G$ is said to be $p$ -hypercyclically embedded in $G$ if every $p$ - $G$ -chief factor of $E$ is cyclic, where $p$ is a fixed prime. In this paper, we prove that $E$ is $p$ -hypercyclically embedded in $G$ if and only if for some $p$ -subgroups $H$ of $E$ , $H \cap X$ have $Π$ -property in $G$ .

Communicated by M. L. Lewis

Keywords:

AMSC: 20D10, 20D15, 20D20

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