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College Algebra Exam Review 88

College Algebra Exam Review 88 - 98 2 BASIC THEORY OF...

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98 2. BASIC THEORY OF GROUPS 2OE4Ł D OE8Ł , 3OE4Ł D OE12Ł , 4OE4Ł D OE2Ł , 5OE4Ł D OE6Ł , 6OE4Ł D OE10Ł , 7OE4Ł D OE0Ł . So the order of OE4Ł is 7. Example 2.2.19. What is the order of OE5Ł in ˚.14/ ? We have OE5Ł 2 D OE11Ł , OE5Ł 3 D OE13Ł , OE5Ł 4 D OE9Ł , OE5Ł 5 D OE3Ł , OE5Ł 6 D OE1Ł , so the order of OE5Ł is 6. Note that j ˚.14/ j D '.14/ D 6 , so this computation shows that ˚.14/ is cyclic, with generator OE5Ł . The next result says that cyclic groups are completely classified by their order. That is, any two cyclic groups of the same order are isomor- phic. Proposition 2.2.20. Let a be an element of a group G . (a) If a has infinite order, then h a i is isomorphic to Z . (b) If a has finite order n , then h a i is isomorphic to the group Z n . Proof. For part (a), define a map ' W Z ! h a i by '.k/ D a k . This map is surjective by definition of h a i , and it is injective because all powers of a are distinct. Furthermore, '.k C l/ D a k C l D a k a l . So ' is an isomorphism between Z and h a i . For part (b), since Z n has n elements OE0Ł; OE1Ł; OE2Ł; : : : ; OEn and h a i has n elements e; a; a 2 ; : : : ; a n 1 , we can define a bijection ' W Z n ! h a i by '.OEkŁ/ D a k for 0 k n 1 . The product (addition) in
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