Finite Groups with Some $c_p$-Supplemented Subgroups
Received:February 24, 2022  Revised:June 26, 2022
Key Words: $c$-supplemented subgroup   $c_p$-normal subgroup   $c_p$-supplemented subgroup   $CS_p$-group   $p$-supersolvable group  
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.12071092) and the Science and Technology Program of Guangzhou Municipality (Grant No.201804010088).
Author NameAffiliation
Yubo LV School of Mathematical Sciences, Guizhou Normal University, Guizhou 550001, P. R. China 
Yangming LI School of Mathematics, Guangdong University of Education, Guangdong 510310, P. R. China 
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      Let $H$ be a subgroup of a finite group $G$ and $p$ a prime divisor dividing the order of $G$. We say $H$ is $c_p$-supplemented in $G$ if there exists a supplement $T$ to $H$ in $G$ containing $H_G$ such that $H\cap T/H_G$ is a $p'$-group, where $H_G$ is the core of $H$ in $G$. A $CS_p$-group is a group in which every $p$-subgroup is $c_p$-supplemented. In this paper, we characterize the $p$-solvability and $p$-supersolvability of groups $G$ with some certain $p$-subgroups being $c_p$-supplemented. Furthermore, we give some equivalent conditions of $CS_p$-group in $p$-solvable universe. Finally, we give some criteria of $CS_p$-groups for the direct product of two $CS_p$-groups. Our results extend some recent conclusions.
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