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A subgroup of a finite group is said to be -supplemented in if there exists a subgroup of such that and is -permutable in . 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 is solvable if every subgroup of odd prime order of is -supplemented in , and that is solvable if and only if every Sylow subgroup of odd order of is -supplemented in . These results improve and extend recent and classical results in the literature.
Lu, Jiakuan, and Qiu, Yanyan. "On solvability of finite groups with some $ss$-supplemented subgroups." Czechoslovak Mathematical Journal 65.2 (2015): 427-433. <http://eudml.org/doc/270115>.
@article{Lu2015, abstract = {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.}, author = {Lu, Jiakuan, Qiu, Yanyan}, journal = {Czechoslovak Mathematical Journal}, keywords = {$ss$-supplemented subgroup; solvable group; supersolvable group}, language = {eng}, number = {2}, pages = {427-433}, publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic}, title = {On solvability of finite groups with some $ss$-supplemented subgroups}, url = {http://eudml.org/doc/270115}, volume = {65}, year = {2015}, }
TY - JOUR AU - Lu, Jiakuan AU - Qiu, Yanyan TI - On solvability of finite groups with some $ss$-supplemented subgroups JO - Czechoslovak Mathematical Journal PY - 2015 PB - Institute of Mathematics, Academy of Sciences of the Czech Republic VL - 65 IS - 2 SP - 427 EP - 433 AB - 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. LA - eng KW - $ss$-supplemented subgroup; solvable group; supersolvable group UR - http://eudml.org/doc/270115 ER -
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