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Given a random cycle, how many conjugates of the cycle can be given? This is, given $\sigma$ how many $\alpha, \beta$ are such that $\alpha\sigma \alpha^{-1}=\beta.$

It is possible to calculate a conjugates, I found posts here that do so: Conjugate permutations in $S_n$ and / or $A_n$.

But is it possible to determine how many are in general? Theorem 5.2 here http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/conjclass.pdf states that

All cycles of the same length in $S_n$ are conjugates

Then to determinate how many conjugates a cycle have, is to know how many conjugates cycles of the same length can be constructed. But how many are there? Aren't $\frac{n!}{r(n-r)!}$ r-cycles in $S_n$? I don't think this gives the number of conjugate cycles, but just r-cycles in general.

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Every $r$-cycle is conjugate to every other $r$-cycle. This holds not just for all $r$, but for more complicated cycle structures. So in $S_4$ we have that $(12)(34)$ is conjugate to $(13)(24)$ because they both have 2 $2$-cycles. Thus to count the size of conjugacy classes, you do as you did to just count the number of such permutations based on cycle structure.

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  • $\begingroup$ Sometimes the word cycle type is used in place of cycle structure. $\endgroup$ – Yacoub Kureh May 10 '18 at 0:51
  • $\begingroup$ So for each cycle there are $(r-1)$ conjugates to it? Then if there are $\frac{n!}{r(n-r)!}$ cycles there should be $\frac{n!(r-1)}{r(n-r)!}$ conjugates. $\endgroup$ – Cure May 10 '18 at 0:53

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