College Algebra Exam Review 133

College Algebra Exam Review 133 - 2.7. QUOTIENT GROUPS AND...

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2.7. QUOTIENT GROUPS AND HOMOMORPHISM THEOREMS 143 Proof. The statement is the special case of the Proposition with G D G=K and ' W G ! G=K the quotient map. Notice that applying the homomor- phism theorem again gives us the isomorphism .G=N/=.K=N/ Š G=K: n Example 2.7.16. What are all the subgroups of Z n ? Since Z n D Z =n Z , the subgroups of Z n correspond one to one with subgroups of Z containing the kernel of the quotient map ' W Z ! Z =n Z , namely n Z . But the subgroups of Z are cyclic and of the form k Z for some k 2 Z . So when does k Z contain n Z ? Precisely when n 2 k Z , or when k divides n . Thus the subgroups of Z n correspond one to one with positive integer divisors of n . The image of k Z in Z n is cyclic with generator ŒkŁ and with order n=k . Example 2.7.17. When is there a surjective homomorphism from one cyclic group Z k to another cyclic group Z ` ? Suppose first that W Z k ! Z ` is a surjective homomorphism such that Œ1Ł D Œ1Ł . Let ' k and ' ` be the natural quotient maps of
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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.

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