A family of non-isomorphism results

TL;DR

This paper proves that groups mV and nV are not isomorphic for any positive integers m, n, or countably infinite, resolving a question posed by Brin.

Contribution

It provides a concise proof demonstrating the non-isomorphism of mV and nV groups for all relevant values, answering an open question.

Findings

mV and nV groups are not isomorphic for all positive integers m, n.

The proof is short and elegant.

Resolves a previously open problem posed by Brin.

Abstract

We give a short argument showing that if $m, n \in {1, 2, ...} \cup {\omega}$, then the groups mV and nV are not isomorphic. This answers a question of Brin.