Algebra

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Math 150a: Modern Algebra Homework 9 Solutions GK1. For which n is the dihedral group D n equiconjugate with O ( 2 ) ? Solution: Let x , y D n be conjugate in O ( 2 ) . Then both x and y are rotations or reflections. If x and y are both rotations, then they must both rotate by the same angle (up to sign). Then either x = y , or x = y 1 . The solution to GK2 on homework #5 covers the conjugacy classes of D n . And, in either case, x and y are conjugate in D n . If x and y are both reflections, then they might not be conjugate in D n if n is even. But they have to be conjugate in D n when n is odd. Hence D n is equiconjugate with O ( 2 ) exactly when n is odd. s GK2. In class I discussed Theorem 5.2.2 from the book, which says that every element of Isom ( R 2 ) is either the identity, a rotation, a reflection, a translation, or a glide reflection. a. Express each of these types of isometries as a product of at most three reflections. Solution: Let φ be a rotation about a point p R 2 by an angle θ . Now let l 1 and l 2 be two lines through p , so that the angle between l 1 and l 2 (measured counterclockwise from l 1 to l 2 ) is equal to / 2. Then, if f 1 and f 2 are the reflections through l 1 and l 2 (respectively); then = f 2 f 1 . If is a reflection of R 2 , it is clearly the product of at most three reflections. p / 2 l 1 l 2 f 1 f 2 (a rotation by angle ) b b b b (a translation of distance t ) t −→ a t / 2 f 1 f 2 b b b −→ a t / 2 (a glide reflection) l b b b b Next, let be the translation of R 2 p m→ p + a , for some vector a with lengh | a | = t . Now let l 1 and l 2
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This homework help was uploaded on 02/01/2008 for the course MATH 150A taught by Professor Kuperberg during the Spring '03 term at UC Davis.

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SOLUTION 9 - Math 150a: Modern Algebra Homework 9 Solutions...

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