Unformatted text preview: disjoint cycles and factorization into a product of transpositions. Compute sign ( σ ) and σ1 . Problem 4. (35 points) For a permutation σ ∈ S n , deﬁne  σ  to be the least integer r > 0 such that σ r = (1). (  σ  is called the order of σ in S n .) a . If σ = σ 1 σ 2 . . . σ k is a product of disjoint cycles, show that  σ  = lcm { σ 1  ,  σ 2  , . . . ,  σ k } . b * . Suppose that σ ∈ S n has k 1 cycles of length 1, k 2 cycles of length 2, k 3 cycles of length 3, . . . , k r cycles of length r , so that n = k 1 + 2 k 2 + 3 k 3 + . . . + rk r . Find  σ  ....
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This note was uploaded on 06/07/2010 for the course MATH 201 taught by Professor Crissinger during the Spring '08 term at University of Delaware.
 Spring '08
 CRISSINGER
 Remainder, Congruence

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