The automorphism group of certain polycyclic groups

TL;DR

This paper fully describes the automorphism groups of certain polycyclic groups called Macdonald groups, detailing their structure, automorphism counts, and conditions for isomorphism, with special cases for specific parameters.

Contribution

It provides a complete characterization of automorphism groups of Macdonald groups, including explicit formulas, embeddings, and conditions for isomorphism, extending previous work on related automorphism structures.

Findings

Automorphism group size for $eta eq 1$ is $2(eta-1)^4$.

Explicit embedding of automorphism groups into ${ m GL}_4(Z/(eta-1)Z)$.

Conditions for isomorphism between $G(eta)$ and $G(eta')$.

Abstract

For $\beta\in{\mathbb Z}$, let $G(\beta)=\langle A,B\,|\, A^{[A,B]}=A,\, B^{[B,A]}=B^\beta\rangle$ be the infinite Macdonald group, and set $C=[A,B]$. Then $G(\beta)$ is a nilpotent polycyclic group of the form $\langle A\rangle\ltimes\langle B,C\rangle$, where $A$ has infinite order. If $\beta\neq 1$, then $G(\beta)$ is of class 3 and $\langle B,C\rangle$ is a finite metacyclic group of order $|\beta-1|^3$, which is an extension of $C_{(\beta-1)^2}$ by $C_{|\beta-1|}$, split except when $v_2(\beta-1)=1$, while $G(1)$ is the integral Heisenberg group, of class 2 and $\langle B,C\rangle\cong{\mathbb Z}^2$. We give a full description of the automorphism group of $G(\beta)$. If $\beta\neq 1$, then $|\mathrm{Aut}(G(\beta))|=2(\beta-1)^4$ and we exhibit an imbedding $\mathrm{Aut}(G(\beta))\hookrightarrow {\mathrm GL}_4({\mathbb Z}/(\beta-1){\mathbb Z})$, but for the case $\beta\in\{-1,3\}$ when 5 is required instead of 4. When $\beta$ is even the automorphism group of $\langle B,C\rangle$ can be obtained from the work of Bidwell and Curran \cite{BC}, and we indicate which of their automorphisms extend to an automorphism of $G(\beta)$. In general, we give necessary and sufficient conditions for $G(\beta)$ to be isomorphic to $G(\gamma)$. When $\gcd(\beta-1,6)=1$, we determine the automorphism group of $L(\beta)=G(\beta)/\langle A^{\beta-1}\rangle$, which is a relative holomorph of $\langle B,C\rangle$, and $\langle A^{\beta-1}\rangle$ is a characteristic subgroup of $G(\beta)$. The map $\mathrm{Aut}(G(\beta))\to \mathrm{Aut}(L(\beta))$ is injective and $\mathrm{Aut}(L(\beta))$ is an extension of the Heisenberg group over ${\mathbb Z}/(\beta-1){\mathbb Z}$ direct product $C_{\beta-1}$, by the holomorph of $C_{\beta-1}$.