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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.09424 |
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Table of Contents:
- For $β\in{\mathbb Z}$, let $G(β)=\langle A,B\,|\, A^{[A,B]}=A,\, B^{[B,A]}=B^β\rangle$ be the infinite Macdonald group, and set $C=[A,B]$. Then $G(β)$ is a nilpotent polycyclic group of the form $\langle A\rangle\ltimes\langle B,C\rangle$, where $A$ has infinite order. If $β\neq 1$, then $G(β)$ is of class 3 and $\langle B,C\rangle$ is a finite metacyclic group of order $|β-1|^3$, which is an extension of $C_{(β-1)^2}$ by $C_{|β-1|}$, split except when $v_2(β-1)=1$, 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 $G(β)$. If $β\neq 1$, then $|\mathrm{Aut}(G(β))|=2(β-1)^4$ and we exhibit an imbedding $\mathrm{Aut}(G(β))\hookrightarrow {\mathrm GL}_4({\mathbb Z}/(β-1){\mathbb Z})$, but for the case $β\in\{-1,3\}$ when 5 is required instead of 4. When $β$ 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 $G(β)$. In general, we give necessary and sufficient conditions for $G(β)$ to be isomorphic to $G(γ)$. When $\gcd(β-1,6)=1$, we determine the automorphism group of $L(β)=G(β)/\langle A^{β-1}\rangle$, which is a relative holomorph of $\langle B,C\rangle$, and $\langle A^{β-1}\rangle$ is a characteristic subgroup of $G(β)$. The map $\mathrm{Aut}(G(β))\to \mathrm{Aut}(L(β))$ is injective and $\mathrm{Aut}(L(β))$ is an extension of the Heisenberg group over ${\mathbb Z}/(β-1){\mathbb Z}$ direct product $C_{β-1}$, by the holomorph of $C_{β-1}$.