arXiv subjects
Publications and source records attributed to Vaibhav Chhajer.
Let $G$ be a finite group. Let $C(G)$ be the set of cyclic subgroups of $G$, $c(G)=|C(G)|$ and $i(G)=\left|\{x\in G : x^{2}=e\}\right|$. In this article, we classify finite groups with $i(G)=c(G)-r$ for $r\in \{0,1,2\}$ and prove that the range of the function given by $β(G)=\frac{i(G)}{c(G)}$ is dense in $[0,1]$. We also answer an open question posed by Gao and Shen in `Finite groups with many cyclic subgroups'.
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 $r\geq-1$.