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Unformatted text preview: 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 OEk ; the distinct cosets are OE0 D n Z ;OE1 D 1 C n Z ;:::;OEn 1 D n 1 C n Z . The product (sum) of two cosets is OEa C OEb D OEa C b . So the group we called Z n is precisely Z =n Z . The quotient homomorphism Z ! Z n is given by k 7! OEk . 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 '.OEk/ 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...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08