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In page Finitely generated abelian group:
"The fundamental theorem for finitely generated abelian groups was proven by Henri Poincaré in 1900, using a matrix proof (which generalizes to principal ideal domains).[citation needed] This was done in the context of computing the homology of a complex, specifically the Betti number and torsion coefficients of a dimension of the complex, where the Betti number corresponds to the rank of the free part, and the torsion coefficients correspond to the torsion part.[2]
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