non-divisibility for abelian groups

Throughout all groups are abelian. We say a group G is n-divisible if nG = G. If G has no non-zero n-divisible subgroups for all n>1 then we say that G is absolutely non-divisible. In the study of class C consisting   all absolutely non-divisible groups such as G, we come across the sub groups T_p(G) = the sum of all p-divisible subgroups and rad_p(G) = the intersection of all p^nG. The properties of these two subgroups have been studied in detail. For classes D_p consisting all p-divisible groups,  F_p consisting all groups with T_p(G) = 0, we prove that (D_p, F_p) is a torsion theory. The class C is closed uder any direct sums and any direct products. If H and G/H are in C then we show that G is in C. Also it is proved that G is in C if and only if rad_p(G)= 0 for any p, if and only if Hom(cup_pD_p, G) = 0. Finally, a more characterization is given for subgroups of  Q(rational numbers) which belongs to C. Various examples are also persented to illustrate the results.