Title: 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 $\mathscr{F}$ 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.