Discrete Heisenberg group and its automorphism group

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}) $.