Czechoslovak Mathematical Journal, first online, pp. 1-5


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

Tao Xu, Heguo Liu

Received June 8, 2019.   Published online April 22, 2020.

Abstract:  Let $\alpha$ and $\beta$ be automorphisms of a nilpotent $p$-group $G$ of finite rank. Suppose that $\langle(\alpha\beta(g))(\beta\alpha(g))^{-1} g\in G\rangle$ 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 $\beta$. (i) $G$ is finite, then $\Gamma$ is an extension of a $p$-group by an abelian group. (ii) $G$ is infinite, then $\Gamma$ is soluble and abelian-by-finite.
Keywords:  automorphism; nilpotent group; finite rank
Classification MSC:  20F18, 20F28
DOI:  10.21136/CMJ.2020.0262-19

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Affiliations:   Tao Xu, Department of Science, Hebei University of Engineering, No. 19 Taiji Road, Handan, 056038, Hebei, P. R. China, e-mail: gtxutao@163.com; Heguo Liu, Department of Mathematics, Hubei University, No. 368 Youyi Road, Wuhan, 430062, Hubei, P. R. China, e-mail: ghliu@hubu.edu.cn


 
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