Isomorphism testing of groups of cube-free order

TL;DR

This paper presents a polynomial-time algorithm for testing isomorphism between cube-free groups, improving previous bounds, with an implementation in GAP for practical use.

Contribution

The paper introduces a new efficient algorithm for isomorphism testing of cube-free groups, including implementation details and complexity analysis.

Findings

Algorithm runs in polynomial time for permutation groups

Implementation provided in GAP

Significantly improves previous super-polynomial bounds

Abstract

A group $G$ has cube-free order if no prime to the third power divides $|G|$. We describe an algorithm that given two cube-free groups $G$ and $H$ of known order, decides whether $G\cong H$, and, if so, constructs an isomorphism $G\to H$. If the groups are input as permutation groups, then our algorithm runs in time polynomial in the input size, improving on the previous super-polynomial bound. An implementation of our algorithm is provided for the computer algebra system {\sf GAP}.