Abstract Nonsense

Crushing one theorem at a time

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

\text{ }

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

\text{ }

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

\text{ }

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

\text{ }

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

\text{ }

\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)

\text{ }

The conclusion follows. \blacksquare

\text{ }

\text{ }

References:

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

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April 16, 2011 - Posted by | Algebra, Group Theory | , , ,

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