Abstract Nonsense

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. $\text{ }$ 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: $\text{ }$ Theorem (Structure Theorem For Finite Abelian Groups)[Existence]: Let $G$ be a finite group, then there exists primes $p_1,\cdots,p_m$ (not necessarily distinct) and integers $n_1,\cdots,n_m$ such that

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$G\cong\mathbb{Z}_{p_1^{n_1}}\times\cdots\times\mathbb{Z}_{p_m^{n_m}}$

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Proof: By lemma 2 we know that there exists integers $k_1,\cdots,k_\ell$ such that

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$G\cong\mathbb{Z}_{k_1}\times\cdots\times\mathbb{Z}_{k_\ell}$

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Then, by the fundamental theorem of arithmetic we know that there exists primes $p_{1,j},\cdots,p_{m_j,j}$ and integers $n_{1,j},\cdots,n_{m_j,j}$ for each $j=1,\cdots,\ell$ such that $k_j=p_{1,j}^{n_{1,j}}\cdots p_{m_j,j}^{n_{m_j,j}}$, and so by lemma 1 we may then conclude that

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$\displaystyle G\cong\prod_{j=1}^{\ell}\left(\mathbb{Z}_{p_{1,j}^{n_{1,j}}}\times\cdots\times\mathbb{Z}_{p_{m_j,j}^{n_{m_j,j}}}\right)$

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The conclusion follows. $\blacksquare$

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References:

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