Finite groups with mostly involuted cyclic subgroups

TL;DR

This paper classifies finite groups based on the relationship between involutions and cyclic subgroups, and explores the density of the ratio of involutions to cyclic subgroups within the interval [0,1].

Contribution

It provides a classification of finite groups with specific involution-to-cyclic subgroup ratios and analyzes the density of this ratio across all finite groups.

Findings

Finite groups with involution count close to the number of cyclic subgroups are classified.

The ratio of involutions to cyclic subgroups is dense in [0,1].

The paper establishes the possible values of this ratio for finite groups.

Abstract

Let $G$ be a finite group, define $I(G)=\{x\in G : x^{2}=1\}$, $C(G)=$ set of the cyclic subgroups of $G$, $i(G)=|I(G)|$ and $c(G)=|C(G)|$. In this article, we will classify finite groups with $i(G)=c(G)-r$ for $r=0,1,$ and $2$. We also prove that the range of the function given by $\beta(G)=\frac{i(G)}{c(G)}$ is dense in $[0,1]$.