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I am trying to see that two permutations are conjugate exactly when they have the same cycle decomposition. I fail to see that

$$r(i_1,i_2,\dots,i_k)r^{−1}=(r(i_1),r(i_2),\dots,r(i_k))$$ for permutations $r$ in a set $X\ni \{i_p\}^k_0$

Any thoughts would be appreciated.

p.s. I ask this question in a new post as suggested by Mr. Gerry Myerson. Still, I believe it belongs to the above thread.

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Evaluate the permutation function on LHS at $r(i_1), r(i_2)$ etc and verify that it matches the cycle on the RHS. Note that LHS is a composition of 3 functions (permutations). You also have to verify that others are fixed points for that permutation.

Let me verify the value of LHS at the element $r(i_1)$. First applying $r^{-1}$ takes it to $i_1$, which in turn is taken to $i_2$ by the cycle. Now this element is carried to $r(i_2)$ by $r$, the third element in the composition. Now you can proceed this way.

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