- W/Prof Cheryl Praeger AM FAA
- Prof Akos Seress

Mar 2009

Computing in Matrix Groups: Constructive Recognition of Classical Groups

A group algorithm takes a group G as its input and attempts to answer a question about the structure of G. In particular, we might wish to design recursive algorithms: to identify a normal subgroup N of G, and reduce our questions to questions about the subgroup N and the quotient G/N. In doing so we can {\it recognise} G, that is, find an isomorphism between G and a group in which computations are more easily performed. A key part of this procedure is recognising the Simple Classical Groups.

In 2008, Magaard, O'Brien and Seress found an algorithm to constructively recognise the General Linear Group \GL(d,q) and the Special Linear Group \SL(d,q) in degree at most d^2, by considering separately the three cases that can occur. Our main research question is to find a similar algorithm to recognise the other simple Classical Groups (the Orthogonal, Unitary and Symplectic Groups).

This project will provide improvements on, and generalisations of, the analyses of existing algorithms for computing in matrix groups, and design new algorithms for constructive recognition of almost simple groups in moderate dimensions.

This work will provide a key component in the worldwide Matrix Group Recognition Project, which aims to provide fast, reliable algorithms for analysing a group input to a computer as matrices.