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# Homework4Solutions - Homework 4 Solutions Math 332, Spring...

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Homework 4 Solutions Math 332, Spring 2010 Problem 1. Proposition. If n 3 , then every element of A n can be written as a product of one or more 3 -cycles. Proof. Let α A n . Since α is an even permutation, it is possible to express α as the product of an even number of 2-cycles σ 1 1 ,...,σ k k , which we can group into pairs as follows: α = σ 1 τ 1 σ 2 τ 2 ··· σ k τ k = ( σ 1 τ 1 )( σ 2 τ 2 ) ··· ( σ k τ k ) . We claim that each product σ i τ i of two 2-cycles can be expressed as the product of one or more three-cycles. There are three cases. If the 2-cycles σ i and τ i are disjoint, say σ i = ( a b ) and τ i = ( c d ), then σ i τ i = ( a b )( c d ) = ( a c d )( a b d ) . If the 2-cycles σ i and τ i are equal, then σ i τ i = σ i 2 = e = (1 2 3)(1 3 2) . The last possibility is that σ i and τ i have one number in common, say σ i = ( a b ) and τ i = ( a c ). In this case, the product is a single 3-cycle: σ i τ i = ( a b )( a c ) = ( a c b

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Homework4Solutions - Homework 4 Solutions Math 332, Spring...

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