Czechoslovak Mathematical Journal, first online, pp. 1-13
Finite groups with a small number of cyclic subgroups
Hailin Liu, Xiangyu Chen, Shouhong Qiao
Received October 21, 2024. Published online April 9, 2025.
Abstract: A finite group $G$ is called an $m$-cyclic group if it has exactly $m$ cyclic subgroups (including the identity subgroup). For $2\leq m\leq12$, the $m$-cyclic groups have been classified in a series of papers. We push the above research work further to classify the finite 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 $|\pi(G)|\leq2$, and one of the following holds: \begin{itemize} \item[(1)] $|\pi(G)|=1$, $G\cong Q_{32}, Z_{11}\times Z_{11}$ or $ Z_{p^{12}}$ with $p$ a prime. \item[(2)] $|\pi(G)|=2$, $G\cong D_{22}$, ${\rm SL}(2, 3)$, $ Z_{11}: Z_5$, $ Z_7 : Z_8$, $ Z_7 : Z_{27}$, $ Z_5 : Z_{16}$, or $ Z_3 : Z_{32}$. \end{itemize}
Affiliations: Hailin Liu, Xiangyu Chen, School of Science, Jiangxi University of Science and Technology, No.86, Hongqi Ave., Ganzhou 341000, P. R. China, e-mail: liuhailin@jxust.edu.cn, wsk44r@126.com; Shouhong Qiao (corresponding author), School of Mathematics and Statistics, Guangdong University of Technology, 161 Yinglong Road, Tianhe District, Guangzhou 510520, P. R. China, e-mail: qshqsh513@163.com
