The automorphism group of certain polycyclic groups

Abstract

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