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Unformatted text preview: Math 150A - Fall 09 - Homework 9 Selected Solutions (Based on solutions prepared by Jeff Ferreira ) 5.4.1 Prove that a discrete group G consisting of rotations about the origin is cyclic and is generated by where is the smallest angle of rotation in G . Proof. This is very similar to a homework problem from last week. In fact, we can use almost the same proof. To convince yourself that a smallest angle of rotation exists, notice that the fact that G is discrete assures us the if and are in G , then | - | . So this means rotations arent too close together. It follows that there exists a smallest angle . Now to do the proof: Pick an arbitrary G . We may write = k + r where 0 r < . Write = k + r = k r and since G and k = ( ) k G , we must have r G . But this contradicts that was minimal, so we must have r = 0. So we showed = k = ( ) k , so generates G , and G is cyclic. 5.5.2 Prove that (5.4) is an equivalence relation. Proof. (5.4) says: Let S be a G-set. Then we have the relation s s if s = gs for some g G . For reflexivity, s s for all s S since s = (1) s where 1 is the identity in G . For symmetry, suppose s s , where s = gs . Then s s by noticing s = g- 1 s . For transitivity, suppose....
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