Symmetry classes ofpp-waves

The isometry groups admitted by plane-fronted gravitational waves with parallel rays are determined without use of any field equations. New groups with 5, 6, and 7 parameters arise which cannot occur for (nontrivial) exact solutions of Einstein's vacuum field equations. For all 17 possible cases the functional form of the free function in the metric is given. We apply the classification to Einstein-Maxwell fields and also determine, in Riemann-Cartan geometry, the form of the torsion tensor by assuming the vanishing of its Lie derivative with respect to the generators of the isometry groups.