Distinguishing crystallographic groups by their finite quotients

TL;DR

This paper demonstrates that in dimensions up to 4, crystallographic groups can be uniquely identified by their finite quotients using computational methods.

Contribution

It provides the first computational proof that all low-dimensional crystallographic groups are distinguishable by their finite quotients.

Findings

All crystallographic groups in dimensions ≤4 are distinguished by finite quotients.

GAP software effectively differentiates these groups.

The method can potentially be extended to higher dimensions.

Abstract

Using the computer algebra program GAP, we show that all crystallographic groups in dimensions at most 4 are distinguished from each other by their sets of finite quotients.