Find permutations in a symmetric group

Question

In the symmetric group $S_{12}$ we have the permutations:

$$\sigma= \begin{bmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ 8 & 5 & 3 & 12 & 10 & 4& 7 &1 & 6 & 2 & 11 & 9 \end{bmatrix}$$

$$\tau= \begin{bmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ 11 & 7 & 1 & 4 & 5 & 8& 2 &12 & 6 & 10 & 3 & 9 \end{bmatrix}$$

$$\theta= \begin{bmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ 7 & 12 & 11 & 2 & 10 & 8& 4 &5 & 3 & 6 & 9 & 1 \end{bmatrix}$$

Find, if exist, permutations $ρ_1$ and $ρ_2$, so that $\tau = ρ_1\sigmaρ_1^{-1}$ and $\theta=ρ_2\sigma ρ_2^{-1}$.

I think $ρ_2$ doesn't exist but I can't explain why (probably because of the row of the disjoint cycle?). And how can I find the permutation $ρ_1$?

Answer 1

Hint:

Two elements of $S_n$ are conjugate to each other iff they are of the same cycle type, i.e. they have same number of cycles of each length.

Suppose $\alpha,\beta$ are of the same cycle type. Then $$\alpha=(a_1\dots a_{n_1})(b_1\dots b_{n_2})\dots(x_1\dots x_{n_k}) $$ and $$\beta=(a'_1\dots a'_{n_1})(b'_1\dots b'_{n_2})\dots(x'_1\dots x'_{n_k})$$ Define $\gamma\in S_n$ where $a_i\mapsto a'_i ,\dots,x_i\mapsto x'_i$.
Check that $\beta=\gamma\alpha\gamma^{-1}$.