A criterion on the finite p-nilpotent groups
Received:June 01, 2018  Revised:October 20, 2018
Key Word: $p$-nilpotent group, the $k$-th center of a group, s-semipermutable subgroup  
Fund ProjectL:NSFC(11271085) and NSF of Guangdong Province (China) (2015A030313791) and Major Projects in Basic Reearch and Applied Research (Natural Science) of Guangdong Provience (2017KZDXM058)
Author NameAffiliationE-mail
XU XIANGYANG Nanchang Normal University 1043991895@qq.com 
LI YANGMING Guangdong University of Education liyangming@gdei.edu.cn 
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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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