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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- Fall '08
- Algebra, positive integers, positive integer, Cyclic group, OED