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Homework2

# Homework2 - Homework 2 Solutions 1 Suppose Sn is an r-cycle...

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Homework 2 Solutions 1. Suppose α S n is an r -cycle. Let α = ( x 1 x 2 . . . x r ) where x i 6 = x j for i 6 = j . (a) α only permutes the elements x 1 through x r and leaves any other elements from { 1 , . . . n } ﬁxed. Thus in order to show that α r = (1) it suﬃces to show that α r ( x i ) = x i for each 1 i r . To see this observe: x 2 = α ( x 1 ) x 3 = α ( x 2 ) = α 2 ( x 1 ) . . . x r = α ( x r - 1 ) = α r - 1 ( x 1 ) x 1 = α ( x r ) = α r ( x 1 ). Similarly for all i , with 1 i r , x i = α r ( x i ). Thus α r = (1). (b) BWOC suppose k such that 0 < k < r and α k = (1). By part (a) we know x k +1 = α k ( x 1 ). Since α k = (1) by our supposition, this implies that x k +1 = x 1 . Since k < r , 1 < k + 1 r . Thus x 1 = x k +1 for 1 < k + 1 r . CONTRADICTION x i ’s are distinct for 1 i r . 2. Let α = ( x 1 x 2 . . . x r ) be an r -cycle. ( ) Suppose

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Homework2 - Homework 2 Solutions 1 Suppose Sn is an r-cycle...

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