Algebra and Geometry | Part IA, 2003

(a) Write the permutation


as a product of disjoint cycles. Determine its order. Compute its sign.

(b) Elements xx and yy of a group GG are conjugate if there exists a gGg \in G such that gxg1=y.g x g^{-1}=y .

Show that if permutations xx and yy are conjugate, then they have the same sign and the same order. Is the converse true? (Justify your answer with a proof or counterexample.)

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