Algorithms for large degree matrix groups

                         Eamonn O'Brien
                     University of Auckland


A major and ongoing project in computational group theory over the 
past decade is the development of good algorithms for the study of 
matrix groups defined over finite fields.  Most of the resulting 
algorithms are available in Magma.

We motivate the project, in particular identifying the limitations 
to existing approaches.  We discuss some natural questions which 
arise: for example, determining the order of a matrix is closely
related to the hard problem of factorising large integers.

The various new approaches are reasonably effective in constructing
the composition factors of the group. Hence, much of the current
focus is on providing good tools to work effectively with the simple 
groups, particularly groups of Lie type. We will also consider how 
this work offers the possibility to obtain detailed structural 
information.