fundamental theorem of finitely generated abelian groupsFundamentalTheoremOfFinitelyGeneratedAbelianGroups2003-08-25 15:29:312008-05-19 12:10:16Theoremfundamental theorem of finitely generated abelian groupsfinitely generatedabelian group% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Rats}{\mathbb{Q}}\begin{thm}[Fundamental Theorem of Finitely Generated Abelian
Groups]
Let $G$ be a finitely generated abelian group. Then there is a
unique expression of the form: $$G\cong
\Ints^{r}\oplus\Ints/n_1\Ints\oplus\Ints/n_2\Ints\oplus\ldots\oplus\Ints/n_s\Ints$$
for some integers $r,n_i$ satisfying:
$$r\geq 0;\quad \forall i, n_i\geq 2;\quad n_{i+1}\mid n_i\ \text{for }
1\leq i\leq s-1.$$
\end{thm}