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Two permutations are conjugate of each other if and only if they have the same cycle structure.

If I want to find the number of elemnts in $S_{13}$ that are both conjugate to (123)(45) and (12)(34), then is the answer 0? because (12)(34) and (123)(45) have not got the same cycle structure.

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    Yes, the answer is 0; if you had an element that is conjugate to both (123)(45) and (12)(34), then they would be conjugate to each other (conjugate to is an equivalence relation, hence transitive). But they are not conjugate to one another.2011-03-03

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That is correct. For this example, also notice that the two elements you mention have different orders. $(12)(34)$ has order 2 while $(123)(45)$ has order 6.

Given a group $G$ suppose that $a,b\in G$ are conjugate. Then there exists $c\in G$ such that $cac^{-1}=b$. If $a$ has order $n$, that is if $a^n=e$, then $b^n=\left(cac^{-1}\right)^n=cac^{-1}cac^{-1}\cdots cac^{-1}=ca^nc^{-1}=cc^{-1}=e$ so that $\text{ord}(b)\leq \text{ord}(a)$. By symmetry, we also have that $\text{ord}(a)\leq \text{ord}(b)$ and hence $\text{ord}(a)=\text{ord}(b)$.

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    @Jonas: Good point. I edited to add in what you said.2011-03-03
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Yes. Even if you didn't know about cycle structure, conjugate elements have the same order, and those two don't.

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    @awllower: Yes, that is why I posted an answer.2011-03-06