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# hw11 - Exercise 7 For each of the following pairs σ,τ of...

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Modern Algebra 1 Homework 5.2 Spring 2010 Due February 17 Exercise 5. Let G be a group and for a, b G define a b if and only if there is an x G so that xax - 1 = b . Prove that is an equivalence relation on G . Exercise 6. Let n 2 and σ S n . Prove that σ ( a 1 a 2 · · · a k ) σ - 1 = ( σ ( a 1 ) σ ( a 2 ) · · · σ ( a k )). Use this to prove that two elements of S n are conjugate if and only if their cycle decompo- sitions have the same “cycle structure.”
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Unformatted text preview: Exercise 7. For each of the following pairs σ,τ of permutations, ﬁnd a permutation γ so that γσγ-1 = τ . a. σ = (15)(23), τ = (13)(45) b. σ = (123)(46)(78), τ = (135)(27)(48) c. σ = (123)(456), τ = (135)(246) Exercise 8. Compute Z ( Q ). [See homework 4.1 for the deﬁnition of Q .]...
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