| A Criterion on the Finite $p$-Nilpotent Groups |
| Received:June 01, 2018 Revised:November 08, 2018 |
| Key Words:
$p$-nilpotent group $k$-th center of a group $s$-semipermutable subgroup
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| Fund Project:Supported by the National Natural Science Foundation of China (Grant No.11271085), the Major Projects in Basic Research and Applied Research (Natural Science) of Guangdong Province (Grant No.2017KZDXM058), Funds of Guangzhou Science and Technology (Grant No.201804010088) and the Science and Technology Research Foundation of Education Department of Jiangxi Province (Grant No.GJJ171109). |
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| Abstract: |
| Let $G$ be a finite group. Suppose that $H$ is a subgroup of $G$. We say that $H$ is $s$-semipermutable in $G$ if $HG_p = G_pH$ for any Sylow $p$-subgroup $G_p$ of $G$ with $(p, |H|) = 1$, where $p$ is a prime dividing the order of $G$. We give a $p$-nilpotent criterion of $G$ under the hypotheses that some subgroups of $G$ are $s$-semipermutable in $G$. Our result is a generalization of the famous Burnside's $p$-nilpotent criterion. |
| Citation: |
| DOI:10.3770/j.issn:2095-2651.2019.03.004 |
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