Beating the Generator-Enumeration Bound for Solvable-Group Isomorphism

TL;DR

This paper presents a novel algorithm for testing isomorphism of solvable groups that outperforms previous methods, leveraging graph representations of generalized composition series and Luks' algorithm.

Contribution

It extends previous results to solvable groups using Hall's Sylow bases, enabling efficient isomorphism testing even with large composition factors.

Findings

Achieves n^((1/2) log_p n + O(log n / log log n)) time complexity for solvable groups

Generalizes composition series with large factors using Hall's Sylow bases

Provides a method to compute canonical forms of solvable groups efficiently

Abstract

We consider the isomorphism problem for groups specified by their multiplication tables. Until recently, the best published bound for the worst-case was achieved by the n^(log_p n + O(1)) generator-enumeration algorithm. In previous work with Fabian Wagner, we showed an n^((1 / 2) log_p n + O(log n / log log n)) time algorithm for testing isomorphism of p-groups by building graphs with degree bounded by p + O(1) that represent composition series for the groups and applying Luks' algorithm for testing isomorphism of bounded degree graphs. In this work, we extend this improvement to the more general class of solvable groups to obtain an n^((1 / 2) log_p n + O(log n / log log n)) time algorithm. In the case of solvable groups, the composition factors can be large which prevents previous methods from outperforming the generator-enumeration algorithm. Using Hall's theory of Sylow bases, we define a new object that generalizes the notion of a composition series with small factors but exists even when the composition factors are large. By constructing graphs that represent these objects and running Luks' algorithm, we obtain our algorithm for solvable-group isomorphism. We also extend our algorithm to compute canonical forms of solvable groups while retaining the same complexity.