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Conditions on $p$ -subgroups implying $p$ -supersolvability

Xiuyun Guo

Department of Mathematics, Shanghai University, Shanghai 200444, P. R. China

E-mail Address: xyguo@staff.shu.edu.cn

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and

Boru Zhang

Department of Mathematics, Shanghai University, Shanghai 200444, P. R. China

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https://doi.org/10.1142/S0219498817501961Cited by:8 (Source: Crossref)

Abstract

In this note, we use fewer $p$ -subgroups $H$ with the condition $O^{p} (G) \cap H ⊴ O^{p} (G)$ to investigate the structure of finite groups. We prove that for a fixed prime $p$ , a given Sylow $p$ -subgroup $P$ of a finite group $G$ , and a power $d$ of $p$ dividing $| G |$ such that $p^{2} \leq d < | P |$ , if $H \cap O^{p} (G)$ is normal in $O^{p} (G)$ for all non-cyclic normal subgroups $H$ of $P$ with $| H | = d$ , then either $G$ is $p$ -supersoluble or else $| P \cap O^{p} (G) | > d$ . This extends the main result of Guo and Isaacs (Conditions on $p$ -subgroups implying $p$ -nilpotence or $p$ -supersolvability, Arch. Math.105 (2015) 215–222). We also derive some applications of the above result which extend some known results.

Keywords:

AMSC: 20D10, 20D20, 20D40

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