Radicals of abelian groups

Abstract

The descripti4 ,ns of torsion 1 lreories a.nd of torsion vnlurd itlernpotent preradicals of a.bc.lian groups are well known. T l~c descri~~tion of all radicals of abclian groups is far from known and it is tho purpose of this paper to tlemonstrat,e Ilia(. the description of radicals whosc vslacs are torsion groups may he as complicatetl as tho tlescri~~tion of all torsion groups t.l~emsrlvcs. For each nou-rtegatiie integcr n , we coilsidcr singly cog'rieratctl radir;ils which fit betwcer~ the radicals pW+'"+' and pW+7L, showing t,h;t there is a proper class ol' sucl~ radicals forming ;L chain therein. We also exar~li~re oll~er intervals in the lattice of radicals sliowing either gaps or thc existence of singly cogencratrd ratliritls in these intervals.