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polycyclic group; regular automorphism; surjectivity
In this paper, we study the structure of polycyclic groups admitting an automorphism of order four on the basis of Neumann's result, and prove that if $\alpha $ is an automorphism of order four of a polycyclic group $G$ and the map $\varphi \colon G\rightarrow G$ defined by $g^{\varphi }=[g,\alpha ]$ is surjective, then $G$ contains a characteristic subgroup $H$ of finite index such that the second derived subgroup $H''$ is included in the centre of $H$ and $C_{H}(\alpha ^{2})$ is abelian, both $C_{G}(\alpha ^{2})$ and $G/[G,\alpha ^{2}]$ are abelian-by-finite. These results extend recent and classical results in the literature.
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