A result on the number of cyclic subgroups of a finite group

Abstract

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 $\alpha (G)\le \alpha (Z(G))$ 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.