These are the (torsion) <a href="http://eom.springer.de/C/c110460.htm">cotorsion groups</a>. By a theorem of Baer all torsion abelian groups that are cotorsion are direct sums of a divisible group and a group of bounded order.

The original paper of R. Baer is "The subgroup of the elements of finite order of an abelian group", Ann. of Math. 37 (1936), 766-781.