Simple Abelian Groups: Can They Be Classified?

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I've been doing some work with simple Abelian groups and their generators, and I feel like there is a way to classify all of them, is this possible?
 
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Yep. It's been done. Since any subgroup of a abelian group is normal, a simple abelian group must have only {e} and itself as subgroups. Thus the only simple abelian groups are the (cyclic) groups of order p, where p is a prime.
 

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