Complexity and computation in matrix groups

The material presented in this chapter comprises an exposition of collaborative research which at the time of writing is still unpublished. Firstly, we discuss a statistical analysis of an algorithm for random selection of elements from groups by Adrian J. Baddeley, Charles R. Leedham-Green, Martin Firth and the first author. Next we describe an improved version of the Meataxe algorithm by Peter M. Neumann and the second author for testing irreducibility of matrix groups and algebras, Finally, we report on a black box algorithm for recognising finite alternating and symmetric groups designed by Robert Beals, Akos Seress, Charles R. Leedham-Green and both authors, We are grateful to our coauthors for their willingness for this work to be included here.