Group isomorphism is nearly-linear time for most orders

TL;DR

This paper demonstrates that group isomorphism can be decided efficiently for most group orders, with nearly-linear time algorithms for dense sets of orders, significantly improving over previous complexities.

Contribution

The authors introduce nearly-linear time algorithms for group isomorphism testing for most orders and for verifying group structure, advancing the computational complexity understanding.

Findings

Decidable group isomorphism in $O(n^2( ext{log } n)^c)$ time for a dense set of orders.

Decidable whether a multiplication table describes a group in $O(n^2( ext{log } n)^c)$ time.

Improved complexity bounds over previous $n^{O( ext{log } n)}$ and $O(n^3)$ algorithms.

Abstract

We show that there is a dense set $\ourset\subseteq \mathbb{N}$ of group orders and a constant $c$ such that for every $n\in \ourset$ we can decide in time $O(n^2(\log n)^c)$ whether two $n\times n$ multiplication tables describe isomorphic groups of order $n$. This improves significantly over the general $n^{O(\log n)}$-time complexity and shows that group isomorphism can be tested efficiently for almost all group orders $n$. We also show that in time $O(n^2 (\log n)^c)$ it can be decided whether an $n\times n$ multiplication table describes a group; this improves over the known $O(n^3)$ complexity. Our complexities are calculated for a deterministic multi-tape Turing machine model. We give the implications to a RAM model in the promise hierarchy as well.