Consider the following statements for countable Abelian groups. There is a direct summand K of G and H such that every direct summand of G and H is embeddable into K. 3. In every infinite decreasing chain of groups, one group is embeddable in a later group. Here we mean literal subgroup, not just up to isomorphism. Reduced means no divisible subgroup.
Torsion group means every element is of finite order.
This is also true for any specific prime p. 1,4,5 are false for reduced torsion groups. 1,4,5 are false for torsion groups. Actually, we need a small refinement of Shore for 5, which looks straightforward.