A new criterion for finite non-cyclic groups
| dc.creator | Zhou, Wei | |
| dc.creator | Shi, Wujie | |
| dc.creator | Duan, Zeyong | |
| dc.date | 2005-09-19 | |
| dc.date.accessioned | 2026-07-07T06:18:11Z | |
| dc.date.available | 2026-07-07T06:18:11Z | |
| dc.description | Let $H$ be a subgroup of a group $G$. We say that $H$ satisfies the power condition with respect to $G$, or $H$ is a power subgroup of $G$, if there exists a non-negative integer $m$ such that $H=G^{m}=<g^{m} | g \in G >$. In this note, the following theorem is proved: Let $G$ be a group and $k$ the number of non-power subgroups of $G$. Then (1) $k=0$ if and only if $G$ is a cyclic group(theorem of F. Sz$\acute{a}$sz) ;(2) $0 < k <\infty$ if and only if $G$ is a finite non-cyclic group; (3) $k=\infty$ if and only if $G$ is a infinte non-cyclic group. Thus we get a new criterion for the finite non-cyclic groups. | |
| dc.description | 6 pages | |
| dc.identifier | https://arxiv.org/abs/math/0509421 | |
| dc.identifier | http://arxiv.org/abs/math/0509421 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/94651 | |
| dc.subject | Group Theory | |
| dc.subject | 20E07, 20E34, 20D25 | |
| dc.title | A new criterion for finite non-cyclic groups | |
| dc.type | text |