I was able to prove the lemma that lets $G$ be a finitely generated abelian group, generated by $n$ elements $\{g_1,g_2,\dotsc,g_n\}$. Then the homomorphism $: \mathbb Z^n \to G$ defined by $(a_1,a_2,\dotsc,a_n) \mapsto g_1^{a_n} g_2^{a_n}$ is an epimorphism. And I understand the kernel of the epimorphism however I am not sure how to use this lemma and the first isomorphism theorem to prove the fundamental theorem of finitely generated abelian groups. Any suggestions as to the right direction would be helpful.
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2Do you mean the homomorphism $(a_1,a_2, \ldots, a_n) \mapsto g_1^{a_1}g_2^{a_2}\cdots g_n^{a_n}$? – David Mehrle Mar 14 '14 at 23:04
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When you say you understand the kernel, does that mean you've realized it can be written as an injective homomorphism $\mathbf{Z}^m \to \mathbf{Z}^n$ (for some $m \leq n$)? (the proof I have in mind would make use of this; maybe other proofs wouldn't) – Mar 14 '14 at 23:17
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dmehrle, Yes that what I meant to put. Thank you – user135524 Mar 17 '14 at 22:50
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Hurkyl, yes. What proof do you have in mind? Thank you – user135524 Mar 17 '14 at 22:53