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College Algebra Exam Review 125

College Algebra Exam Review 125 - 2.7 QUOTIENT GROUPS AND...

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2.7. QUOTIENT GROUPS AND HOMOMORPHISM THEOREMS 135 Example 2.7.2. (Finite cyclic groups as quotients of Z ). The construction of Z n in Section 1.7 is an example of the quotient group construction. The (normal) subgroup in the construction is n Z D f `n W ` 2 Z g . The cosets of n Z in Z are of the form k C n Z D OEkŁ ; the distinct cosets are OE0Ł D n Z ; OE1Ł D 1 C n Z ; : : : ; OEn D n 1 C n Z . The product (sum) of two cosets is OEaŁ C OEbŁ D OEa C . So the group we called Z n is precisely Z =n Z . The quotient homomorphism Z ! Z n is given by k 7! OEkŁ . Example 2.7.3. Now consider a cyclic group G of order n with generator a . There is a homomorphism ' W Z ! G of Z onto G defined by '.k/ D a k . The kernel of this homomorphism is precisely all multiples of n , the order of a ; ker .'/ D n Z . I claim that ' “induces” an isomorphism Q ' W Z n ! G , defined by Q '.OEkŁ/ D a k D '.k/ . It is necessary to check that this makes sense (i.e., that Q ' is well defined) because we have attempted to define the value of Q ' on a coset OEkŁ
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