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# h5 - 1 Homework V 2.25(i If is an r-cycle show that r =(1...

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1 Homework V: July 10, 2009 2 . 25 ( i ) . If α is an r -cycle, show that α r = (1). Write α = ( i 0 i 1 . . . i r - 1 ). The proof of Lemma 2.25(ii) shows that α k ( i 0 ) = i k , where subscripts are read mod r . It follows that α r ( i 0 ) = i 0 . But we know that we can also write α = ( i j i j +1 . . . i j - 1 ) for any j , so that α r ( i j ) = i j . Therefore, α r fixes all i j . Since α fixes any other integers between 1 and n , if any, so does α r . Therefore, α r = (1), for it fixes everything. 2 . 25 ( ii ) . If α is an r -cycle, show that r is the least positive integer such that α r = (1). If 0 < k < r , then α k ( i 0 ) = i k = i 0 , and so α k = (1). 2 . 30 ( i ) . Prove that if α, β are (not necessarily disjoint) permutations that com- mute, then ( αβ ) k = α k β k for all k 1. We prove first, by induction on n 1, that βα n = α n β . The base step n = 1 is true because we are assuming that α and β commute. For the inductive step, βα n +1 = βα n α = α n βα = α n αβ = α n +1 β (the second equality is the inductive hypothesis). We now prove the desired result by induction on k 1. The base step k = 1 is obviously true. For the inductive step, ( αβ ) k +1 = αβ ( αβ ) k = αβα

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