I'm actually going to answer a slightly different question, which is the one I think I really wanted to ask: is this question motivated by category theory? I first saw this question on an old qualifying exam, and it seemed like there was more to it. In fact, I was told last week that it's related to the idea of localization. A good source of information on this concept can be found here (it wouldn't let me direct link to the pdf, but it's the paper number 101 called localizations on that page). The qualifying exam question can now be phrased: abelian groups are closed under localization. The same question has been posed for other classes of groups. For example, finite groups, perfect groups and torsion abelian groups are all not closed under localization. Nilpotent groups of class 2 are closed under localization.