On groups of automorphisms of nilpotent $p$-groups of finite rank

Xu, Tao ; Liu, Heguo

Czechoslovak Mathematical Journal, Tome 70 (2020) no. 4, pp. 1161-1165.

Résumé

Let $\alpha $ and $\beta $ be automorphisms of a nilpotent $p$-group $G$ of finite rank. Suppose that $\langle (\alpha \beta (g))(\beta \alpha (g))^{-1}\colon g\in G\rangle $ is a finite cyclic subgroup of $G$, then, exclusively, one of the following statements holds for $G$ and $\Gamma $, where $\Gamma $ is the group generated by $\alpha $ and $\beta $. \item {(i)} $G$ is finite, then $\Gamma $ is an extension of a $p$-group by an abelian group. \item {(ii)} $G$ is infinite, then $\Gamma $ is soluble and abelian-by-finite.

Détail

DOI : 10.21136/CMJ.2020.0262-19

Classification : 20F18, 20F28
Keywords: automorphism; nilpotent group; finite rank