Adv Alegbra HW Solutions 46

Adv Alegbra HW Solutions 46 - 46 (ii) If kr n , where 1 < r...

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46 (ii) If kr n , where 1 < r n , prove that the number of permuta- tions α S n , where α is a product of k disjoint r -cycles, is 1 k ! 1 r k [ n ( n 1 ) ··· ( n + 1 ). ] Solution. Absent. 2.25 (i) If α is an r -cycle, show that α r = ( 1 ) . Solution. If α = ( i 0 ... i r 1 ) , then the proof of Lemma 2.25(ii) shows that α k ( i 0 ) = i k , where the subscript is read mod r . Hence, α r ( i 0 ) = i 0 . But the same is true if we choose the notation for α having any of the other i j as the f rst entry. (ii) If α is an r -cycle, show that r is the least positive integer k such that α k = ( 1 ) . Solution. Use Proposition 2.24. If k < r , then α k ( i 0 ) = i k ±= i 0 , so that α k ±= 1. 2.26 Show that an r -cycle is an even permutation if and only if r is odd. Solution. In the proof of Proposition 2.35, we showed that any r -cycle α is a product of r 1 transpositions. The result now follows from Proposi- tion 2.39, for sgn (α) = ( 1 ) r 1 =− 1. 2.27 Given X ={ 1 , 2 ,..., n } , let us call a permutation τ of X an adjacency if it is a transposition of the form ( ii + 1 ) for i < n .I f i < j , prove that ( ij ) is a product of an odd number of adjacencies.
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.

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