Finite Groups with Some $c_p$-Supplemented Subgroups

Abstract:

      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.