On the number of cyclic subgroups of a finite group

TL;DR

This paper investigates the ratio of cyclic subgroups to group order in finite groups, analyzing its properties, extremal cases, and connections to commutation probability, with classifications for specific group families.

Contribution

It introduces a detailed study of the function lpha(G) = c(G)/|G|, characterizes groups with maximal lpha(G), and explores its behavior in direct powers and quotient groups.

Findings

Characterized groups with maximal lpha(G) in various families

Classified groups where lpha(G) > 3/4

Derived asymptotic results for direct powers

Abstract

Let $G$ be a finite group and let $c(G)$ be the number of cyclic subgroups of $G$. We study the function $\alpha(G) = c(G)/|G|$. We explore its basic properties and we point out a connection with the probability of commutation. For many families $\mathscr{F}$ of groups we characterize the groups $G \in \mathscr{F}$ for which $\alpha(G)$ is maximal and we classify the groups $G$ for which $\alpha(G) > 3/4$. We also study the number of cyclic subgroups of a direct power of a given group deducing an asymptotic result and we characterize the equality $\alpha(G) = \alpha(G/N)$ when $G/N$ is a symmetric group.