4023f11ps8a

4023f11ps8a - Homework #8 Solutions Due: November 9, 2011...

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Unformatted text preview: Homework #8 Solutions Due: November 9, 2011 1. Factor each of the following permutations into disjoint cycles. Using your answer express the inverse as a product of disjoint cycles and determine if each permutation is even or odd. (a) 1 2 3 4 5 6 7 8 9 4 7 9 8 2 1 6 3 5 (b) 1 2 3 4 5 6 7 8 9 6 4 8 9 3 1 7 5 2 (c) 1 3 2 5 7 3 8 5 (d) 3 5 7 2 3 4 7 6 3 5 7 1 I Solution. (a) = 1 2 3 4 5 6 7 8 9 4 7 9 8 2 1 6 3 5 = 1 4 8 3 9 5 2 7 6 Thus is a 9-cycle and hence 1 = 6 7 2 5 9 3 8 4 1 . Since 9 is odd, is an even permutation. (b) = 1 2 3 4 5 6 7 8 9 6 4 8 9 3 1 7 5 2 = 1 6 2 4 9 3 8 5 . Hence, 1 = 1 6 2 9 4 3 5 8 and is odd since it is a product of an odd permuta- tion and two even permutations. J 2. (a) How many permutations in S 5 flx 1? I Solution. Fixing 1 means that a permutation becomes a permutation on the set f 2 ; 3 ; 4 ; 5 g and there are 4! = 24 such permutations. J (b) How many permutations in...
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4023f11ps8a - Homework #8 Solutions Due: November 9, 2011...

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