number of elements in $S_{13}$ conjugate to both (12)(34) and (123)(45)

Question

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.

Answer 1

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 $\text{ord}(a)=\text{ord}(b)$.

Answer 2

Yes. Even if you didn't know about cycle structure, conjugate elements have the same order, and those two don't.