Finitely Generated Torsion-Free Nilpotent Groups Admitting an Automorphism of Prime Order

Abstract

Let $G$ be a finitely generated torsion-free nilpotent group and $α$ an automorphism of prime order $p$ of $G$. If the map $φ : G → G$ defined by $g^φ = [g, α]$ is surjective, then the nilpotent class of $G$ is at most $h(p)$, where $h(p)$ is a function depending only on $p$. In particular, if $α^3 = 1$, then the nilpotent class of $G$ is at most $2$.