Assignment11Part2

Assignment11Part2 - Math 302 Modern Algebra Spring 2009...

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Unformatted text preview: Math 302 Modern Algebra Spring 2009 Homework Solutions 7.9 THE SYMMETRIC AND ALTERNATING GROUPS 3. Express as a product of disjoint cycles. (a) 1 2 3 4 5 6 7 8 9 2 1 3 5 4 7 9 8 6 = (1 2)(3)(4 5)(6 7 9)(8) (b) 1 2 3 4 5 6 7 8 9 3 5 1 2 4 6 8 9 7 = (1 3)(2 5 4)(6)(7 8 9) (d) (1 4)(2 7)(5 2 3)(3 4)(1 4 7 2) = (1 5 7 3)(2 4) (e) (7 2 3 6)(8 5)(5 7 1)(1 5 3 7)(4 8 6) = (1 2 3)(4 5 6)(7 8) 4. Write each permutation in Exercise 3 as a product of transpositions. (a) 1 2 3 4 5 6 7 8 9 2 1 3 5 4 7 9 8 6 = (1 2)(4 5)(6 7 9) = (1 2)(4 5)(6 7)(7 9) (b) 1 2 3 4 5 6 7 8 9 3 5 1 2 4 6 8 9 7 = (1 3)(2 5 4)(7 8 9) = (1 3)(2 5)(5 4)(7 8)(8 9) (d) (1 4)(2 7)(5 2 3)(3 4)(1 4 7 2) = (1 5 7 3)(2 4) = (1 5)(5 7)(7 3)(2 4) (e) (7 2 3 6)(8 5)(5 7 1)(1 5 3 7)(4 8 6) = (1 2 3)(4 5 6)(7 8) = (1 2)(2 3)(4 5)(5 6)(7 8) 6. List the elements in each group. (c) A 4 S 4 has 4! = 24 elements, including 2-cycles, 3-cycles, 4-cycles, and products of disjoint 2- cycles. S 4 = { (1) , (1 2) , (1 3) , (1 4) , (2 3) , (2 4) , (3 4) , (1 2 3) , (1 2 4) , (1 3 2) , (1 3 4) , (1 4 2) , (1 4 3) , (2 3 4) , (2 4 3) , (1 2 3 4) , (1 2 4 3) , (1 3 2 4) , (1 3 4 2) , (1 4 2 3) , (1 4 3 2) , (1 2)(3 4) , (1 3)(2 4) , (1 4)(2 3) } . The 3-cycles can be written as products of two transpositions, so are even. The 4-cycles can be written as products of three transpositions, so are odd. Then A 4 contains the three cycles and the products of disjoint 2-cycles: A 4 =...
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Assignment11Part2 - Math 302 Modern Algebra Spring 2009...

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