Polycyclic groups with automorphisms of order four

Xu, Tao ; Zhou, Fang ; Liu, Heguo

Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 575-582.

Résumé

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 $\phi :G\to G$ defined by $g^{\phi }=[g,\alpha ]$ 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.

Détail

DOI : 10.1007/s10587-016-0276-8

Classification : 20E36
Mots-clés : polycyclic group; regular automorphism; surjectivity