Computing in Matrix Groups

                            Mark Stather
                         University of Warwick


The aim of the Matrix Groups Recognition project is to develop a range of
algorithms for matrix groups similar to those available for permutation
groups.  I will describe an algorithm that uses Aschbacher's Theorem on
the structure of subgroups of GL(d,q) to compute a chief series for a
given matrix group G, thus avoiding BSGS methods. This chief series can
then be reordered to exhibit R:=Soluble Radical, the Socle of G module R
and the kernel of the action of G on the Soc(G/R).  We are then able to
apply the tecniques developed in the permutation group case of first solving a
problem in G/R then lifting the solution down through the elementary abelian
layers in the soluble radical.  In particular this applies to computing all
the normal subgroups of G, centralisers of elements and subgroups in G and the
conjugacy classes of G.  I will also describe an easy application of this
method to compute a Sylow subgroup of a matrix group.