College Algebra Exam Review 193

College Algebra Exam Review 193 - 3.6. FINITELY GENERATED...

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Unformatted text preview: 3.6. FINITELY GENERATED ABELIAN GROUPS 203 that 1 D t 1 r 1 C t 2 r 2 C C t s r s . (For the computation of the integers t 1 ;t 2 ;:::;t s , see Example 3.5.11 .) Thus for any x 2 Z , x D xt 1 r 1 C xt 2 r 2 C C xt s r s . Taking residues mod n , OEx D xt 1 OEr 1 C xt 2 OEr 2 C C xt s OEr s . This is the decomposition of OEx with components in the subgroups G i . Example 3.6.16. Consider the primary decomposition of Z 60 , Z 60 D GOE2 GOE3 GOE5; where GOE2 is the unique subgroup of Z 60 of size 4, namely GOE2 D h OE15 i ; GOE3 is the unique subgroup of Z 60 of size 3, namely GOE3 D h OE20 i ; and GOE5 is the unique subgroup of Z 60 of size 5, namely GOE5 D h OE12 i . We can compute integers t 1 ;t 2 , and t 3 satisfying t 1 15 C t 2 20 C t 3 12 D 1 , namely . 5/15 C .5/20 C . 2/12 D 1 . Therefore, for any integer x , OEx D 5xOE15 C 5xOE20 2xOE12 . This gives us the unique decomposition of OEx as a sum OEx D a 2 C a 3 C a 5 , where a j 2 GOEj...
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