Geodesic
Polycyclic groups with automorphisms of order four
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 575-582 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.1007/s10587-016-0276-8
Classification : 20E36
Keywords: polycyclic group; regular automorphism; surjectivity 
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     title = {Polycyclic groups with automorphisms of order four},
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     pages = {575--582},
     year = {2016},
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Xu, Tao; Zhou, Fang; Liu, Heguo. Polycyclic groups with automorphisms of order four. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 575-582. doi: 10.1007/s10587-016-0276-8

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