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Unformatted text preview: a 3 C t 4 a 4 . 3.6. Finitely generated abelian groups In this section, we obtain a structure theorem for nitely generated abelian groups. The theorem states that any nitely generated abelian group is a direct product of cyclic groups, with each factor either of in-nite order or of order a power of a prime; furthermore, the number of the cyclic subgroups appearing in the direct product decomposition, and their orders, are unique. Two nite abelian groups are isomorphic if, and only if, they have the same decomposition into a direct product of cyclic groups of innite or prime power order. The Invariant Factor Decomposition Every nite abelian group G is a quotient of Z n for some n . In fact, if x 1 ;:::;x n is a set of generators of minimum cardinality, we can dene a homomorphism of abelian groups from Z n onto G by '. P i r i O e i / D...
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- Fall '08