Finite groups with nearly half as many cyclic subgroups as elements

Abstract

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 $2\le r\le |G|-1$. 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 $q\ne p)$, or $|C(G)|=\frac{|G|}{2}+2^k,k\ge 0$. Then, we partially answer the open question by classifying finite groups $G$ having $\frac{|G|}{2}-1\le |C(G)|\le \frac{|G|}{2}+1$ 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 $r\ge -1$.

12 pages