# A Polycyclic Presentation for the q-Tensor Square of a Polycyclic Group

## TL;DR

This paper develops a polycyclic presentation for the q-tensor square of polycyclic groups, extending existing methods for q=0, and provides tools for computing the q-exterior centre and related homology groups.

## Contribution

It introduces a new polycyclic presentation for the q-tensor square of polycyclic groups and extends prior methods to all q ≥ 0.

## Key findings

- Derived a polycyclic presentation for G ⊗^q G.
- Presented presentations for G ∧^q G and H_2(G, Z_q).
- Established a criterion for computing the q-exterior centre.

## Abstract

Let $G$ be a group and $q$ a non-negative integer. We denote by $\nu^q(G)$ a certain extension of the $q$-tensor square $G \otimes^q G$ by $G \times G$. In this paper we derive a polycyclic presentation for $G \otimes^q G$, when $G$ is polycyclic, via its embedding into $\nu^q(G)$. Furthermore, we derive presentations for the $q$-exterior square $G \wedge^q G$ and for the second homology group $H_2(G, \mathbb{Z}_q).$ Additionally, we establish a criterion for computing the $q-$exterior centre $Z_q^\wedge (G)$ of a polycyclic group $G, $ which is helpful for deciding whether $G$ is capable modulo $q$. These results extend to all $q \geq 0$ existing methods due to Eick and Nickel for the case $q = 0$.