College Algebra Exam Review 88

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

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Unformatted text preview: 98 2. BASIC THEORY OF GROUPS 2Œ4 D Œ8, 3Œ4 D Œ12, 4Œ4 D Œ2, 5Œ4 D Œ6, 6Œ4 D Œ10, 7Œ4 D Œ0. So the order of Œ4 is 7. Example 2.2.19. What is the order of Œ5 in ˚.14/? We have Œ52 D Œ11, Œ53 D Œ13, Œ54 D Œ9, Œ55 D Œ3, Œ56 D Œ1, so the order of Œ5 is 6. Note that j˚.14/j D '.14/ D 6, so this computation shows that ˚.14/ is cyclic, with generator Œ5. The next result says that cyclic groups are completely classified by their order. That is, any two cyclic groups of the same order are isomorphic. Proposition 2.2.20. Let a be an element of a group G . (a) If a has infinite order, then hai is isomorphic to Z. (b) If a has finite order n, then hai is isomorphic to the group Zn . Proof. For part (a), define a map ' W Z ! hai by '.k/ D ak . This map is surjective by definition of hai, and it is injective because all powers of a are distinct. Furthermore, '.k C l/ D ak Cl D ak al . So ' is an isomorphism between Z and hai. For part (b), since Zn has n elements Œ0; Œ1; Œ2; : : : ; Œn 1 and hai has n elements e; a; a2 ; : : : ; an 1 , we can define a bijection ' W Zn ! hai by '.Œk/ D ak for 0 Ä k Ä n 1. The product (addition) in Zn is given by Œk C Œl D Œr, where r is the remainder after division of k C l by n. The multiplication in hai is given by the analogous rule: ak al D ak Cl D ar , where r is the remainder after division of k C l by n. Therefore, ' is an isomorphism. I Subgroups of Cyclic Groups In this subsection, we determine all subgroups of cyclic groups. Since every cyclic group is isomorphic either to Z or to Zn for some n, it suffices to determine the subgroups of Z and of Zn . Proposition 2.2.21. (a) Let H be a subgroup of Z. Then either H D f0g, or there is a unique d 2 N such that H D hd i D d Z. (b) If d 2 N , then d Z Š Z. (c) If a; b 2 N , then aZ  b Z if, and only if, b divides a. Proof. Let’s check part (c) first. If aZ  b Z, then a 2 b Z, so b divides a. On the other hand, if b ja, then a 2 b Z, so aZ  b Z. ...
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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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