Article
| Title: | Finite groups with a small number of cyclic subgroups (English) |
| Author: | Liu, Hailin |
| Author: | Chen, Xiangyu |
| Author: | Qiao, Shouhong |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 75 |
| Issue: | 3 |
| Year: | 2025 |
| Pages: | 839-851 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | A finite group $G$ is called an $m$-cyclic group if it has exactly $m$ cyclic subgroups (including the identity subgroup). For $2\leq m\leq 12$, the $m$-cyclic groups have been classified in a series of papers. We push the above research work further to classify the finite \hbox {13-cyclic} groups, which could be considered as a step to answer the open problem posed by M. Tărnăuceanu (2015). The detailed structure of many groups of ``small'' orders is also analyzed. The following main theorem is proved: Let $G$ be a finite 13-cyclic group. Then $|\pi (G)|\leq 2$, and one of the following holds: \begin {itemize} \item [(1)] $|\pi (G)|=1$, $G\cong Q_{32},{\bf Z}_{11}\times {\bf Z}_{11}$ or ${\bf Z}_{p^{12}}$ with $p$ a prime. \item [(2)] $|\pi (G)|=2$, $G\cong D_{22}$, ${\rm SL}(2, 3)$, ${\bf Z}_{11}: {\bf Z}_{5}$, ${\bf Z}_7 : {\bf Z}_8$, ${\bf Z}_7 : {\bf Z}_{27}$, ${\bf Z}_5 : {\bf Z}_{16}$, or ${\bf Z}_3 : {\bf Z}_{32}$. \end {itemize} (English) |
| Keyword: | finite group |
| Keyword: | $m$-cyclic group |
| Keyword: | cyclic subgroup |
| MSC: | 20D20 |
| MSC: | 20D25 |
| DOI: | 10.21136/CMJ.2025.0441-24 |
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| Date available: | 2025-09-19T11:53:10Z |
| Last updated: | 2025-09-22 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153055 |
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