College Algebra Exam Review 63

College Algebra Exam Review 63 - 1.10 GROUPS 73 First by...

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Unformatted text preview: 1.10. GROUPS 73 First, by hypothesis OEaŁ has a multiplicative inverse OExŁ and OEbŁ has a multiplicative inverse OEyŁ . Then OEaŁOEbŁ has multiplicative inverse OEyŁOExŁ , because .OEaŁOEbŁ/.OEyŁOExŁ/ D OEaŁ.OEbŁ.OEyŁOExŁ// D OEaŁ..OEbŁOEyŁ/OExŁ/ D OEaŁ.OE1ŁOExŁ/ D OEaŁOExŁ D OE1Ł: A second way to see that OEabŁ 2 ˚.n/ is that the invertibility of OEaŁ and OEbŁ implies that a and b are relatively prime to n , and it follows that ab is also relatively prime to n . Hence OEabŁ 2 ˚.n/ . It is clear that OE1Ł 2 ˚.n/ . Now since multiplication is associative on Z n , it is also associative on ˚.n/ , and since OE1Ł is a multiplicative identity for Z n , it is also a multiplicative identity for ˚.n/ . Finally, every element in ˚.n/ has a multiplicative inverse, by definition of ˚.n/ . This proves that ˚.n/ is a group. n Now we state a basic theorem about finite groups, whose proof will have to wait until the next chapter (Theorem 2.5.6 ): If G is a finite group of size...
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