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I'm trying to study the different cycle types and their conjugates in a symmetric group $S_n$.

For more of a concrete example, suppose we are given a cycle $\sigma \in S_5$. My questions are:

How to determine what elements are conjugate to $\sigma$,

how many elements are of the same cycle type as $\sigma$,

lastly how to determine what the elements of the centralizer $C_G(\sigma)$ are.

Any help would be great!

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Let $\sigma$ be a cycle of order $m$ in $S_n$

  1. The elements that are conjugate to $\sigma$ are also cycle of order $m$.

  2. The number of elements that are of same cycle type of $\sigma$ is $(1/m)(n)(n-1)\dots(n-m+1)$

  3. $|C_G(\sigma)|=(n-m)!m$

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  • $\begingroup$ What if $\sigma = (12)(34)$, or any disjoint product of two cycles? Does the formula above still apply? $\endgroup$ – Sank Nov 17 '16 at 2:22
  • $\begingroup$ @user282639 No, those properties are only for cycle. $\endgroup$ – Alan Wang Nov 17 '16 at 2:27
  • $\begingroup$ Then how would I go about determining the 3 things for a non r-cycle? i.e. $(12)(34), (123)(4657)$ etc... $\endgroup$ – Sank Nov 17 '16 at 2:29
  • $\begingroup$ Then the elements that are conjugate must have same cycle structure. For example the elements which are conjugate to $(12)(34)$ are $(13)(24),(14)(23)$ $\endgroup$ – Alan Wang Nov 17 '16 at 2:32
  • $\begingroup$ How can I determine how many elements have such cycle types? $\endgroup$ – Sank Nov 17 '16 at 4:05

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