Finite groups with nearly half as many cyclic subgroups as elements
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916884664811520 |
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| author | Chhajer, Vaibhav Hatui, Sumana Sharma, Palash |
| author_facet | Chhajer, Vaibhav Hatui, Sumana Sharma, Palash |
| contents | Suppose $C(G)$ denotes the set of all cyclic subgroups of a finite group $G$, and $\mathcal{O}_{2}(G)$ denotes the number of elements of order $2$ in $G$. In [Marius T., Finite groups with a certain number of cyclic subgroups. The American Mathematical Monthly 122.3 (2015): 275-276], an open problem was asked to classify the groups $G$ with $|C(G)|=|G|-r$, where $2 \leq r \leq |G|-1$. In this article, first we show that, for an odd prime $p$, there are infinitely many groups $G$ with $|C(G)|= \frac{|G|}{2}$, $|C(G)|=\frac{|G|}{p^{q-1}}$ (for prime $q\neq p)$, or $|C(G)|=\frac{|G|}{2}+2^{k}, k\geq 0$. Then, we partially answer the open question by classifying finite groups $G$ having $\frac{|G|}{2}-1\leq |C(G)| \leq \frac{|G|}{2}+1$ for some fix values of $\mathcal{O}_{2}(G)$. Finally, we provide a complete list of finite groups $G$ having $|C(G)|=\frac{|G|+(2r+1)}{2}$ for $r\geq-1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_21163 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finite groups with nearly half as many cyclic subgroups as elements Chhajer, Vaibhav Hatui, Sumana Sharma, Palash Group Theory 20D25, 20E34 Suppose $C(G)$ denotes the set of all cyclic subgroups of a finite group $G$, and $\mathcal{O}_{2}(G)$ denotes the number of elements of order $2$ in $G$. In [Marius T., Finite groups with a certain number of cyclic subgroups. The American Mathematical Monthly 122.3 (2015): 275-276], an open problem was asked to classify the groups $G$ with $|C(G)|=|G|-r$, where $2 \leq r \leq |G|-1$. In this article, first we show that, for an odd prime $p$, there are infinitely many groups $G$ with $|C(G)|= \frac{|G|}{2}$, $|C(G)|=\frac{|G|}{p^{q-1}}$ (for prime $q\neq p)$, or $|C(G)|=\frac{|G|}{2}+2^{k}, k\geq 0$. Then, we partially answer the open question by classifying finite groups $G$ having $\frac{|G|}{2}-1\leq |C(G)| \leq \frac{|G|}{2}+1$ for some fix values of $\mathcal{O}_{2}(G)$. Finally, we provide a complete list of finite groups $G$ having $|C(G)|=\frac{|G|+(2r+1)}{2}$ for $r\geq-1$. |
| title | Finite groups with nearly half as many cyclic subgroups as elements |
| topic | Group Theory 20D25, 20E34 |
| url | https://arxiv.org/abs/2506.21163 |