Unformatted text preview: (i) Prove that if α and β are (not necessarily disjoint) permutations that commute, then (αβ) k = α k β k for all k ≥ 1. Solution. We prove f rst, by induction on k ≥ 1, that βα k = α k β . The base step is true because α and β commute. For the inductive step, βα k + 1 = βα k α = α k βα ( inductive hypothesis ) = α k αβ = α k + 1 β. We now prove the result by induction on k ≥ 1. The base step is obviously true. For the inductive step, (αβ) k + 1 = αβ(αβ) k = αβα k β k ( inductive hypothesis ) = αα k ββ k ( proof above ) = α k + 1 β k + 1 ....
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 Fall '11
 KeithCornell
 Mathematical Induction, Inductive Reasoning, Solution., Structural induction, inductive hypothesis, regular permutations

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