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 ﬁnitely generated abelian groups. The theorem states that any ﬁnitely generated abelian group is a direct product of cyclic groups, with each factor either of in-ﬁnite 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 ﬁnite abelian groups are isomorphic if, and only if, they have the same decomposition into a direct product of cyclic groups of inﬁnite or prime power order. The Invariant Factor Decomposition Every ﬁnite 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 deﬁne 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
- Algebra, Abelian group, Cyclic group, free abelian group, Zm, abelian groups