Constructive homomorphisms for classical groups

TL;DR

This paper develops efficient algorithms for computing projections and coset representatives in classical groups, facilitating group recognition and conjugacy problems in computational group theory.

Contribution

It introduces polynomial-time algorithms for projecting normalisers, representing quotient groups with few generators, and constructing isometries, advancing computational methods for classical groups.

Findings

Polynomial-time algorithms for projections and coset representatives

Representation of quotient groups with minimal generators and relations

Fast construction of isometries between spaces with forms

Abstract

Let Omega be a quasisimple classical group in its natural representation over a finite vector space V, and let Delta be its normaliser in the general linear group. We construct the projection from Delta to Delta/Omega and provide fast, polynomial-time algorithms for computing the image of an element. Given a discrete logarithm oracle, we also represent Delta/Omega as a group with at most 3 generators and 6 relations. We then compute canonical representatives for the cosets of Omega. A key ingredient of our algorithms is a new, asymptotically fast method for constructing isometries between spaces with forms. Our results are useful for the matrix group recognition project, can be used to solve element conjugacy problems, and can improve algorithms to construct maximal subgroups.