1.I.1B

(a) Write the permutation

$(123)(234)$

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

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

Show that if permutations $x$ and $y$ 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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