Computing with Group Homomorphisms

Let G = (X) and H be finite groups and let r : X ~ H be a map from the generating set X of G into H. We describe a simple approach for deciding whether or not r determines a group homomorphism from G to H, and if it does, for computing the kernel of r If O and H are permutation groups the algorithm is a simple application of standard algorithms for bases and strong generating sets. If G and H are soluble groups given in the usual way by PAG-systems and corresponding power conjugate presentations then the algorithm is a simple application of the non-commutative Gauss algorithm for constructing a subgroup of a soluble group. Further, a probabilistic algorithm is given for finding the kernel and image of 4~ when each of G and H is given as a permutation group or a soluble group, IGI is known, and r is known to determine a homomorphism.