Finite groups with nearly half as many cyclic subgroups as elements

TL;DR

This paper investigates the structure of finite groups with a number of cyclic subgroups close to half the group size, providing classifications and infinite examples for certain subgroup counts.

Contribution

It offers new classifications of finite groups with specific ratios of cyclic subgroups to group order, addressing an open problem and expanding known examples.

Findings

Infinite groups with specific cyclic subgroup ratios identified

Partial classification for groups with subgroup counts near half the order

Complete list of groups with a particular cyclic subgroup to order ratio

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 $2 \leq r \leq |G|-1$. 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 $r\geq-1$.