For any radical R of abelian groups which does not commute with arbitrary cartesian products we define the norm kRk to be the least cardinal for which there exists a family, of this size, of groups G a such that R G a j RG a . This norm kRk is always regular. Assuming GCH, we construct reduced products G to show that every regular cardinal k which is not greater that any weakly compact cardinal is the norm of a suitable group radical R G .