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ProblemSet2 - α and β for which αβ 2 6 = α 2 β 2...

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Homework 2 Due: Wednesday 1/27/10 Problems after the double bar will not be collected but it is recommended that you do them. My assumption is that you will do them. 1. (Problem 2.25) (a) If α is an r -cycle, show that α r = (1) (b) If α is an r -cycle, show that r is the least positive integer k such that α k = (1). ( r is called the order of α ). 2. (Problem 2.26) Show that an r -cycle is an even permutation if and only if r is odd. 3. (Problem 2.30) (a) Prove that if α and β are (not necessarily disjoint) permutations that commute, then ( αβ ) k = α k β k for all k 1. (b) Give an example of 2 permutations
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Unformatted text preview: α and β for which ( αβ ) 2 6 = α 2 β 2 . 4. (Problem 2.32) If n ≥ 2, prove that the number of even permutations in S n is 1 2 n !. 5. (Problem 2.34) If n ≥ 3, show that if α ∈ S n commutes with every β ∈ S n , then α = (1). 6. Let α be the following permutation in S 9 ± 1 2 3 4 5 6 7 8 9 9 4 7 6 5 3 8 2 1 ² (a) Write α as a product of disjoint cycles. (b) Is α an even permutation or an odd permutation? Write it as a product of transpositions. (c) Find α-1 , the inverse of α ....
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