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

Authors: libo Zhao, yangming li, LU GONG

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.

Keywords: $p$-group of maximal class, extra-special $p$-group

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