On the number of cyclic subgroups of a finite group

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 $\mathcal{F}$ of groups we characterize the groups $G\in \mathcal{F}$ 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.

Cited by in corpus (1)

• Maximal covers of finite groups