On the finite $p$-groups with unique cyclic subgroup of given order

Abstract

In this paper, we prove that if $G$ is nonabelian and $|G|>p^4$, then $G$ has a unique cyclic subgroup of order $p^m$ with $m\geq 3$ if and only if $G$ has a unique abelian subgroup of order $p^3$ if and only if $G$ is a $2$-group of maximal class.