3.5. LINEAR ALGEBRA OVER Z 193 Proof. Condition (a) trivially implies condition (b). If (b) holds, then each standard basis element O e j is in Z f v 1 ;:::;v n g , O e j D X i a i;j v i : (3.5.1) Let A D .a i;j / . The n equations 3.5.1 are equivalent to the single matrix equation E D PA , where E is the n –by– n identity matrix. Thus, P is invertible with inverse A 2 Mat n . Z / . If P is invertible with inverse A 2 Mat n . Z / , then E D PA implies that each standard basis element O e j is in Z f v 1 ;:::;v n g , so f v 1 ;:::;v n g generates Z n . Moreover, ker .P/ D f0 g means that f v 1 ;:::;v n g is linearly independent. n We can now combine Proposition 3.5.9 and Lemma 3.5.12 to obtain our main result about bases of subgroups of Z n . Theorem 3.5.13. If N is a subgroup of Z n , then N is a free abelian group of rank s ± n . Moreover, there exists a basis f v 1 ;:::;v n g of Z n , and there exist positive integers d 1 ;d 2 ;:::;d s , such that d i divides d j if i ± j and f d 1 v 1
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