College Algebra Exam Review 179

College Algebra Exam Review 179 - 3.5 LINEAR ALGEBRA OVER Z...

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Unformatted text preview: 3.5. LINEAR ALGEBRA OVER Z 189 Corollary 3.5.8. Every subgroup of a finitely generated abelian group is finitely generated. Proof. Let G be a finitely generated abelian group, with a generating set f x 1 ;:::;x n g . Define homomorphism from Z n onto G by '. P i r i O e i / D P i r i x i . Let A be a subgroup of G and let N D ' 1 .A/ . According to the previous lemma, N has a generating set X with no more than n elements. Then '.X/ is a generating set for A with no more than n elements. n We know that if N is an s dimensional subspace of the vector space K n , then there is a basis f v 1 ;:::;v n g of K n such that f v 1 ;:::;v s g is a basis of N . For subgroups of Z n , the analogous statement is the following: If N is a nonzero subgroup of Z n , then there exist a basis f v 1 ;:::;v n g of Z n , s 1 , and nonzero elements d 1 ;d 2 ;:::;d s of Z , with d i divid- ing d j if i j such that f d 1 v 1 ;:::;d s v s g is a basis of N . In particular, N is free....
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