## Review of Group Theory: The Structure Theorem for Finite Abelian Groups[Almost](Pt. II)

**Point of Post: **This post is a continuation of this one. Putting these two lemmas together gives us the uniqueness part of the structure theorem also known affectionately as the Fundamental Theorem for Finitely Generated Abelian Groups (F.T.F. Gag). Indeed: **Theorem (Structure Theorem For Finite Abelian Groups)[Existence]:** *Let be a finite group, then there exists primes (not necessarily distinct) and integers such that*

**Proof: **By lemma 2 we know that there exists integers such that

Then, by the fundamental theorem of arithmetic we know that there exists primes and integers for each such that , and so by lemma 1 we may then conclude that

The conclusion follows.

**References:**

1. Simon, Barry. *Representations of Finite and Compact Groups*. Providence, RI: American Math. Soc., 1996. Print.

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