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Unformatted text preview: Homework #7 Due: October 18, 2010 Do the following exercises from Lax: Page 124: 9.l, 9.3, 9.5. Do the following exercises from the Burnside Lemma supplement: Page 235: 6, 7, 12. This supplement has many worked out examples that you may study for how to apply Burnsides Lemma. 9.1. a) Find the number of difierent squares with vertices colored red, white, or blue. b) Find the number of difierent mcolored squares for any m . I Solution. The symmetry group of a square is D 4 = ' ; fi; fi 2 ; fi 3 ; fl; fifl; fi 2 fl; fi 3 fl and the set X on which we want D 4 = G to act is the set of all functions from the vertices of the square to the set of m colors. Identifying the elements of D 4 with permutations of the vertices of the square, as described on page 114, we can compute the cycle type of each permutation determined by g 2 D 4 and then compute j X g j using Theorem 5.5.5, page 231 of the supplement (or corollary 10.3, page 127 of Lax). The results are, for mcolors: Cycle Rep. l ( ) j X j (1)(2)(3)(4) 4 m 4 fi (1 2 3 4) 1 m fi 2 (1 3)(2 4) 2 m 2 fi 3 (1 4 3 2) 1 m fl (1 2)(3 4) 2 m 2 fifl (1 3)(2)(4) 3 m 3 fi 2 fl (1 ; 4)(2 ; 3) 2 m 2 fi 3 fl (1)(2 4)(3) 3 m 3 Hence, Burnsides Lemma give the number of orbits as N = 1 8 ( m 4 + 2 m 3 + 3 m 2 + 2 m ) : This is the answer to part b). To get part a), let m = 3 to get N = 1 8 (81 + 54 + 27 + 6) = 168 8 = 21 : J 9.3. Find the number of difierent regular hexagons with vertices colored red or blue....
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 Fall '09

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