mar14 - Math 4124 Monday, March 14 March 14, Ungraded...

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Unformatted text preview: Math 4124 Monday, March 14 March 14, Ungraded Homework Exercise 4.2.1 on page 121 Let G = { 1 , a , b , c } be the Klein 4-group. (a) Label 1 , a , b , c with the integers 1,2,4,3, respectively, and prove that under the left regular representation of G into S 4 the nonidentity elements are mapped as follows: a 7 ( 1 2 )( 3 4 ) b 7 ( 1 4 )( 2 3 ) c 7 ( 1 3 )( 2 4 ) (b) Relabel 1 , a , b , c as 1,4,2,3, respectively, and compute the image of each element of G under the left regular representation of G into S 4 . Show that the image of G in S 4 under this labelling is the same subgroup as the image of G in part (a) (even though the nonidentity elements individually map to different permutations under the two different labellings). Of course in both cases 1 gets mapped to the identity permutation (1). (a) a sends 1 to a 1 = a , a to aa = 1, b to ab = c , c to ac = b . Therefore a corresponds to the permutation (1 2)(3 4). b sends 1 to b 1 = b , a to ba = c , b to bb = 1, c to bc = a . Therefore b corresponds to the permutation (1 4)(2 3) c sends 1 to c 1 = c , a to ca = b , b to cb = a , c to cc = 1. Therefore c corresponds to the permutation (1 3)(2 4) (b) a sends 1 to a 1 = a , a to aa = 1, b to ab = c , c to ac = b . Therefore a corresponds to the permutation (1 4)(2 3). b sends 1 to b 1 = b , a to ba = c , b to bb = 1, c to bc = a . Therefore b corresponds to the permutation (1 2)(3 4) c sends 1 to c 1 = c , a to ca = b , b to cb = a , c to cc = 1. Therefore c corresponds to the permutation (1 3)(2 4) In both cases, the image of G in S 4 is { 1, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3) } . Exercise 4.3.2 on page 130 Find all the conjugacy classes and their sizes in the following groups. (a) D 8 (b) Q 8 (c) A 4 (a) We write D 8 = h r , s | r 4 = s 2 = 1 , rs = sr- 1 i . Then the conjugacy classes are { 1 } , { r 2 } , { r , r 3 } , { s , sr 2 } , { sr , sr 3 } . For example to do the conjugacy class containing s , we have 1 s 1- 1 = s , rsr- 1 = sr- 2 = sr 2 so the conjugacy class has size at least 2. On the other hand { 1 , s , r 2 , sr...
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mar14 - Math 4124 Monday, March 14 March 14, Ungraded...

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