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College Algebra Exam Review 193

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

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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 , OExŁ D xt 1 OEr 1 Ł C xt 2 OEr 2 Ł C C xt s OEr s Ł . This is the decomposition of OExŁ with components in the subgroups G i . Example 3.6.16. Consider the primary decomposition of Z 60 , Z 60 D GOE2Ł GOE3Ł GOE5Ł; where GOE2Ł is the unique subgroup of Z 60 of size 4, namely GOE2Ł D h OE15Ł i ; GOE3Ł is the unique subgroup of Z 60 of size 3, namely GOE3Ł D h OE20Ł i ; and GOE5Ł is the unique subgroup of Z 60 of size 5, namely GOE5Ł D h OE12Ł 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 , OExŁ D 5xOE15Ł C 5xOE20Ł 2xOE12Ł . This gives us the unique decomposition of OExŁ as a sum OExŁ D a 2 C a 3 C a 5 , where a j 2 GOEj Ł . For example, OE13Ł D 65OE15Ł C 65OE20Ł 26OE12Ł D 3OE15Ł C 2OE20Ł C 4OE12Ł .
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