A characterization of an abelian group Let $G$ be an abelian group. Is there a characterization of $G$ whenever every subgroup of $G$ is a direct summand of $G$?
 A: Such groups are exactly the groups that are a (possibly infinite) direct sum of cyclic groups of prime order.  Here is a sketch of the proof.  First, given a direct sum of cyclic groups of prime order, any subgroup can be canonically decomposed into its $p$-torsion for different primes $p$, and now you use the fact that any subspace of an $\mathbb{F}_p$-vector space is a direct summand.  Conversely, if every subgroup of $G$ is a direct summand, pick a nonzero element $g\in G$; it is not hard to see that the subgroup $\mathbb{Z}g$ generated by $g$ must be finite.  There is then a subgroup $P$ of $\mathbb{Z}g$ that is cyclic of prime order, which is a summand of $G$ by hypothesis; write $G=P\oplus H$.  We can now repeat this argument to split off another summand from $H$ which is cyclic of prime order, and we can continue splitting off such summands by transfinite induction until we have exhausted all of $G$.
Another characterization of such groups is that they are exactly the (abelian) groups in which every element is torsion with squarefree order.  Indeed, any torsion abelian group decomposes as the direct sum of its $p$-power torsion over all primes $p$, and the squarefree condition then says that the $p$-power torsion is always an $\mathbb{F}_p$-vector space.
More generally, a module over a ring which has the property that every submodule is a direct summand is called semisimple.  An argument similar to the one sketched above shows a module is semisimple iff it is a direct sum of simple modules, where a simple module is a module with no nontrivial submodules.
