Discrete Heisenberg group and its automorphism group

TL;DR

This paper provides a simpler, more transparent proof that the automorphism group of the discrete Heisenberg group ${ m Heis}(3, bZ)$ is isomorphic to $(bZ imes bZ) times GL(2,bZ)$, clarifying its structure.

Contribution

It offers an easier, more transparent proof of the automorphism group's structure for the discrete Heisenberg group, improving understanding of its algebraic properties.

Findings

Automorphism group is isomorphic to $(bZ imes bZ) times GL(2,bZ)$

Provides a clearer, more accessible proof of the isomorphism

Enhances understanding of the automorphism group's structure

Abstract

In this note we give more easy and short proof of a statement previously proved by P. Kahn that the automorphism group of the discrete Heisenberg group ${\rm Heis}(3, \mathbb{Z}) $ is isomorphic to the group $ (\mathbb{Z} \oplus \mathbb{Z}) \rtimes GL(2,\mathbb{Z})$. The method which we suggest to construct this isomorphism gives far more transparent picture of the structure of the automorphism group of the group ${\rm Heis}(3, \mathbb{Z}) $.