# Incorporating Weisfeiler-Leman into algorithms for group isomorphism

## TL;DR

This paper introduces a novel algorithm for general finite group isomorphism testing by integrating algebraic group techniques with combinatorial Weisfeiler-Leman methods on hypergraphs, enhancing the efficiency and flexibility of GpI algorithms.

## Contribution

It combines multiple advanced GpI algorithms with Weisfeiler-Leman hypergraph techniques, creating a unified, adaptable framework for testing finite group isomorphism.

## Key findings

- Framework effectively encodes low-genus quotients as hypergraphs.
- The approach is flexible to include additional invariants.
- Combines algebraic and combinatorial methods for improved GpI.

## Abstract

In this paper we combine many of the standard and more recent algebraic techniques for testing isomorphism of finite groups (GpI) with combinatorial techniques that have typically been applied to Graph Isomorphism. In particular, we show how to combine several state-of-the-art GpI algorithms for specific group classes into an algorithm for general GpI, namely: composition series isomorphism (Rosenbaum-Wagner, Theoret. Comp. Sci., 2015; Luks, 2015), recursively-refineable filters (Wilson, J. Group Theory, 2013), and low-genus GpI (Brooksbank-Maglione-Wilson, J. Algebra, 2017). Recursively-refineable filters -- a generalization of subgroup series -- form the skeleton of this framework, and we refine our filter by building a hypergraph encoding low-genus quotients, to which we then apply a hypergraph variant of the k-dimensional Weisfeiler-Leman technique. Our technique is flexible enough to readily incorporate additional hypergraph invariants or additional characteristic subgroups.