A Criterion on the Finite $p$-Nilpotent Groups
Received:June 01, 2018  Revised:November 08, 2018
Key Word: $p$-nilpotent group   $k$-th center of a group   $s$-semipermutable subgroup  
Fund ProjectL: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).
Author NameAffiliation
Xiangyang XU Department of Mathematics, Nanchang Normal University, Jiangxi 330032, P. R. China 
Yangming LI Department of Mathematics, Guangdong University of Education, Guangdong 510310, P. R. China 
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      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.
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