On subgroups of totally projective primary abelian groups and direct sums of cyclic groups

For abelian p-groups, a new characterization of C-decomposability is given. It implies, for example, that it is consistent with ZFC that every group whose final rank has countable cofinality is C-decomposable. The class of groups that can be embedded in totally projective groups of length omega . 2 is also considered. It is known that such groups might not be C-decomposable, but it is shown that they possess a related property; that is, they are far from thick.