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College Algebra Exam Review 178

College Algebra Exam Review 178 - 188 3 PRODUCTS OF GROUPS...

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188 3. PRODUCTS OF GROUPS combination of finitely many elements of B . Since S is finite, it is con- tained in the subgroup geneated by a finite subset B 0 of B . But then G D Z S Z B 0 . So B 0 generates G . It follows from the previous lemma that B 0 D B . n Proposition 3.5.5. Any two bases of a finitely generated free abelian group have the same cardinality. Proof. Let G be a finitely generated free abelian group. By the previous lemma, any basis of G is finite. If G has a basis with n elements, then G Š Z n , by Proposition 3.5.2 . Since Z n and Z m are nonisomorphic if m ¤ n , G cannot have bases of different cardinalities. n Definition 3.5.6. The rank of a finitely generated free abelian group is the cardinality of any basis. Proposition 3.5.7. Every subgroup of Z n can be generated by no more than n elements. Proof. The proof goes by induction on n . We know that every subgroup of Z is cyclic (Proposition 2.2.21 ), so this takes care of the base case n D 1 . Suppose that n > 1 and that the assertion holds for subgroups of
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