On solvability of finite groups with some s s -supplemented subgroups

Jiakuan Lu; Yanyan Qiu

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 2, page 427-433
  • ISSN: 0011-4642

Abstract

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A subgroup H of a finite group G is said to be s s -supplemented in G if there exists a subgroup K of G such that G = H K and H ∩ 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 s s -supplemented in G , and that G is solvable if and only if every Sylow subgroup of odd order of G is s s -supplemented in G . These results improve and extend recent and classical results in the literature.

How to cite

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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 -

References

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