An axiomatization for the universal theory of the Heisenberg group

TL;DR

This paper provides a complete axiomatization of the universal theory of the Heisenberg group, showing it is characterized by specific quasi-identities and properties of centralizers within the group.

Contribution

It offers a full proof that the universal theory of the Heisenberg group is axiomatized by quasi-identities and abelian centralizer conditions, extending previous partial results.

Findings

The universal theory of the Heisenberg group is axiomatized by quasi-identities.

Centralizers of noncentral elements in the Heisenberg group are abelian.

The paper completes earlier partial results with a full proof.

Abstract

The Heisenberg group, here denoted $H$, is the group of all $3\times 3$ upper unitriangular matrices with entries in the ring $\mathbb{Z}$ of integers. A.G. Myasnikov posed the question of whether or not the universal theory of $H$, in the language of $H$, is axiomatized, when the models are restricted to $H$-groups, by the quasi-identities true in $H$ together with the assertion that the centralizers of noncentral elements be abelian. Based on earlier published partial results we here give a complete proof of a slightly stronger result.