A note on infinite $aS$-groups

Reza Nikandish; Babak Miraftab

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 4, page 1003-1009
  • ISSN: 0011-4642

Abstract

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Let $G$ be a group. If every nontrivial subgroup of $G$ has a proper supplement, then $G$ is called an $aS$-group. We study some properties of $aS$-groups. For instance, it is shown that a nilpotent group $G$ is an $aS$-group if and only if $G$ is a subdirect product of cyclic groups of prime orders. We prove that if $G$ is an $aS$-group which satisfies the descending chain condition on subgroups, then $G$ is finite. Among other results, we characterize all abelian groups for which every nontrivial quotient group is an $aS$-group. Finally, it is shown that if $G$ is an $aS$-group and $|G|\neq pq,p$, where $p$ and $q$ are primes, then $G$ has a triple factorization.

How to cite

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Nikandish, Reza, and Miraftab, Babak. "A note on infinite $aS$-groups." Czechoslovak Mathematical Journal 65.4 (2015): 1003-1009. <http://eudml.org/doc/276164>.

@article{Nikandish2015,
abstract = {Let $G$ be a group. If every nontrivial subgroup of $G$ has a proper supplement, then $G$ is called an $aS$-group. We study some properties of $aS$-groups. For instance, it is shown that a nilpotent group $G$ is an $aS$-group if and only if $G$ is a subdirect product of cyclic groups of prime orders. We prove that if $G$ is an $aS$-group which satisfies the descending chain condition on subgroups, then $G$ is finite. Among other results, we characterize all abelian groups for which every nontrivial quotient group is an $aS$-group. Finally, it is shown that if $G$ is an $aS$-group and $|G|\neq pq,p$, where $p$ and $q$ are primes, then $G$ has a triple factorization.},
author = {Nikandish, Reza, Miraftab, Babak},
journal = {Czechoslovak Mathematical Journal},
language = {eng},
number = {4},
pages = {1003-1009},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on infinite $aS$-groups},
url = {http://eudml.org/doc/276164},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Nikandish, Reza
AU - Miraftab, Babak
TI - A note on infinite $aS$-groups
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 4
SP - 1003
EP - 1009
AB - Let $G$ be a group. If every nontrivial subgroup of $G$ has a proper supplement, then $G$ is called an $aS$-group. We study some properties of $aS$-groups. For instance, it is shown that a nilpotent group $G$ is an $aS$-group if and only if $G$ is a subdirect product of cyclic groups of prime orders. We prove that if $G$ is an $aS$-group which satisfies the descending chain condition on subgroups, then $G$ is finite. Among other results, we characterize all abelian groups for which every nontrivial quotient group is an $aS$-group. Finally, it is shown that if $G$ is an $aS$-group and $|G|\neq pq,p$, where $p$ and $q$ are primes, then $G$ has a triple factorization.
LA - eng
UR - http://eudml.org/doc/276164
ER -

References

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