Constructing a short defining set of relations for a finite group

An algorithm for the construction of a defining set of relations w.r.t. a given set of generators of a finite group G is presented. Compared with previously known methods it yields fewer relations and is better suited for iterated application to large groups. These improvements are achieved by considering the action of some subgroup H < G on the vertices of the Cayley graph of G w.r.t. the subgroup H.