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* A group is Abelian if its [[commutator subgroup]] is trivial.

==Examples==

[[Cyclic group]]s are good examples of Abelian groups. Further, any direct product of cyclic groups is also an Abelian group. Further, every [[finitely generated group|finitely generated]] Abelian group is obtained this way. This is the famous [[structure theorem for finitely generated Abelian groups]].

The structure theorem can be used to generate a complete listing of finite Abelian groups, as described here: [[classification of finite Abelian groups]].

==Facts==

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