Geodesic
On solvability of finite groups with some $ss$-supplemented subgroups
Czechoslovak Mathematical Journal, Tome 65 (2015) no. 2, pp. 427-433
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A subgroup $H$ of a finite group $G$ is said to be $ss$-supplemented in $G$ if there exists a subgroup $K$ of $G$ such that $G=HK$ and $H\cap K$ is $s$-permutable in $K$. In this paper, we first give an example to show that the conjecture in A. A. Heliel's paper (2014) has negative solutions. Next, we prove that a finite group $G$ is solvable if every subgroup of odd prime order of $G$ is $ss$-supplemented in $G$, and that $G$ is solvable if and only if every Sylow subgroup of odd order of $G$ is $ss$-supplemented in $G$. These results improve and extend recent and classical results in the literature.
A subgroup $H$ of a finite group $G$ is said to be $ss$-supplemented in $G$ if there exists a subgroup $K$ of $G$ such that $G=HK$ and $H\cap K$ is $s$-permutable in $K$. In this paper, we first give an example to show that the conjecture in A. A. Heliel's paper (2014) has negative solutions. Next, we prove that a finite group $G$ is solvable if every subgroup of odd prime order of $G$ is $ss$-supplemented in $G$, and that $G$ is solvable if and only if every Sylow subgroup of odd order of $G$ is $ss$-supplemented in $G$. These results improve and extend recent and classical results in the literature.
DOI : 10.1007/s10587-015-0186-1
Classification : 20D10, 20D20, 20D40
Keywords: $ss$-supplemented subgroup; solvable group; supersolvable group 
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Lu, Jiakuan; Qiu, Yanyan. On solvability of finite groups with some $ss$-supplemented subgroups. Czechoslovak Mathematical Journal, Tome 65 (2015) no. 2, pp. 427-433. doi: 10.1007/s10587-015-0186-1

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