On the number of cyclic subgroups of a finite group

Let $G$ be a finite group and let $c(G)$ be the number of cyclic subgroups of $G$. We study the function $\alpha (G)=c(G)\mathrm{/}\mathrm{\mid }G\mathrm{\mid }$. 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 $G \in \mathscr{F}$forwhich$\alpha(G)$ismaximalandweclassifythegroups$G$forwhich$\alpha(G) > 3/4$. 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 $\alpha (G)=\alpha (G\mathrm{/}N)$ when $G\mathrm{/}N$ is a symmetric group.