Generating Random Elements Of A Finite Group

We present a "practical" algorithm to construct random elements of a finite group. We analyse its theoretical behaviour and prove that asymptotically it produces uniformly distributed tuples of elements. We discuss tests to assess its effectiveness and use these to decide when its results are acceptable for some matrix groups. 1 Introduction How do we select random elements from a large finite group G? Our answer will depend on the type of description we have for the group. If G is a permutation group, we can construct a base and strong generating set; if G is a finitely-presented soluble group, we can obtain a power-conjugate presentation. In both cases, we can use these special descriptions to obtain random elements. Our algorithm is designed for the case where G is described by a generating set X , and we have no convenient canonical form for the elements of G. In particular, we consider the case where G is a subgroup of GL(d; q), the group of non-singular d \Theta d matrices def..