College Algebra Exam Review 89

College Algebra Exam Review 89 - s such that sŒdŁ D Œ0Ł...

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2.2. SUBGROUPS AND CYCLIC GROUPS 99 Next we verify part (a). Let H be a subgroup of Z . If H ¤ f 0 g , then H contains a nonzero integer; since H contains, together with any integer a , its opposite ± a , it follows that H contains a positive integer. Let d be the smallest element of H \ N . I claim that H D d Z . Since d 2 H , it follows that h d i D d Z ² H . On the other hand, let h 2 H , and write h D qd C r , where 0 ³ r < d . Since h 2 H and qd 2 H , we have r D h ± qd 2 H . But since d is the least positive element of H and r < d , we must have r D 0 . Hence h D qd 2 d Z . This shows that H ² d Z . So far we have shown that there is a d 2 N such that d Z D H . If also d 0 2 H and d 0 Z D H , then by part (a), d and d 0 divide one another. Since they are both positive, they are equal. This proves the uniqueness of d in part (a). Finally, for part (b), we can check that a 7! da is an isomorphism from Z onto d Z . n Corollary 2.2.22. Every subgroup of Z other than f 0 g is isomorphic to Z . Lemma 2.2.23. Let n ´ 2 and let d be a positive divisor of n . The cyclic subgroup h ŒdŁ i generated by ŒdŁ in Z n has cardinality jh ŒdŁ ij D n=d . Proof. The order of ŒdŁ is the least positive integer
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Unformatted text preview: s such that sŒdŁ D Œ0Ł , by Proposition 2.2.17 , that is, the least positive integer s such that n divides sd . But this is just n=d , since d divides n . n Proposition 2.2.24. Let H be a subgroup of Z n . (a) Either H D f Œ0Ł g , or there is a d > 0 such that H D h ŒdŁ i . (b) If d is the smallest of positive integers s such that H D h ŒsŁ i , then d j H j D n . Proof. Let H be a subgroup of Z n . If H ¤ f Œ0Ł g , let d denote the smallest of positive integers s such that ŒsŁ 2 H . An argument identical to that used for Proposition 2.2.21 (a) shows that h ŒdŁ i D H . Clearly, d is then also the smallest of positive integers s such that h ŒsŁ i D H . Write n D qd C r , where ³ r < d . Then ŒrŁ D ± qŒdŁ 2 h ŒdŁ i . Since r < d and d is the least of positive integers s such that ŒsŁ 2 h ŒdŁ i ,...
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