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Chapter 2 Review: Solved problems

1. In S10, let = (1,3,5,7,9), = (1,2,6), and = (1,2,5,3). For = , write as a product of disjoint cycles, and use this to find its order and its inverse. Is even or odd?

2. Define the function f : Z17× -> Z17× by f(x) = x-1, for all x in Z17×. Is f one to one? Is f onto? If possible, find the inverse function f-1.

3. (a) Let be a fixed element of Sn. Show that f : Sn -> Sn defined by f() = -1, for all in Sn, is a one-to-one and onto function.

(b) In S3, let = (1,2). Compute the corresponding function f defined in part (a).