  Let $G$ be a finite group and $\alpha(G)=\frac{|C(G)|}{|G|}$\,, where $C(G)$
denotes the set of cyclic subgroups of $G$. In this short note, we prove that
$\alpha(G)\leq\alpha(Z(G))$ and we describe the groups $G$ for which the
equality occurs. This gives some sufficient conditions for a finite group to be
$4$-abelian or abelian.

Group Theory