The descriptions of torsion theories and of torsion valued idempotent preradicals of abclian groups are well known. The description of all radicals of abelian groups is far from known and it is the purpose of this paper to demonstrat that the description of radicals whose values are torsion groups may he as complicated as the description of all torsion groups themselves. For each nou-negative integer n, we consider singly cogenerated radicals which fit between the radicals Pw+n+1and Pw+nshowing that there is a proper class of such radicals forming a chain therein. We also examine other intervals in the lattice of radicals showing either gaps or the existence of singly cogenerated radicals in these intervals.