4023f11ex3

4023f11ex3 - (a) Determine the orbit, Orb( x ) for x = 1...

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Name: Exam 3 Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Put your name on each page of your paper. 1. [15 Points] Let = ± 1 2 3 4 5 6 2 1 3 5 4 6 and = ± 1 2 3 4 5 6 6 1 2 4 3 5 . Compute the following permutations: (a) fifl (b) flfi (c) ¡ 1 2. [12 Points] Let ± 2 S 10 be the permutation ± 1 2 3 4 5 6 7 8 9 10 9 7 8 4 3 2 10 5 1 6 : (a) Find the cycle decomposition of ± . (b) Determine if ± is even or odd. 3. [12 Points] How many permutations in S 7 have cycle structure (1 ; 2 ; 2 ; 2)? 4. [12 Points] What is the number of elements in each of the following groups? (a) S n (c) D n (c) C n 5. [15 Points] Let G be the following group of permutations of the set X = f 1 ; 2 ; 3 ; 4 ; 5 g : G = ' (1) ; ¡ 1 4 ¢¡ 2 5 3 ¢ ; ¡ 2 3 5 ¢ ; ¡ 1 4 ¢ ; ¡ 2 5 3 ¢ ; ¡ 2 3 5 ¢¡ 1 4 ¢“ = ' e; ±; ± 2 ; ± 3 ; ± 4 ; ± 5 : Thus G acts on X via these permutations.
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Unformatted text preview: (a) Determine the orbit, Orb( x ) for x = 1 and x = 2. (b) Determine the stabilizer subgroup Stab( x ) for x = 1 and x = 2. (c) Use your results to verify that j Orb( x ) j = j G j = j Stab( x ) j for x = 1 and x = 2. 6. [9 Points] Fill in the missing parts to complete the statement of Burnside’s Theorem: If G is a flnite group that acts on a flnite set X , then the number k of distinct orbits is: k = 1 X : 7. [25 Points] Find the number of patterns obtained by coloring the vertices of a regular pentagon with 4 colors. Be sure to explain your calculations, including identiflcation of the symmetry group and all of the flxed point calculations needed. Math 4023 November 21, 2011 1...
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This document was uploaded on 12/28/2011.

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