Product of a disjoint cycle

Question

Would somebody help me with part (b) the following question?

I have proven part (a) but have no idea how to do (b).

Thank you!

Answer 1

Hint: Let \begin{eqnarray*} \sigma_1&=&(a_{1,1}a_{1,2} \cdots a_{1,l_1}) (a_{2,1}a_{2,2} \cdots a_{2,l_2}) \cdots (a_{r,1}a_{r,2} \cdots a_{r,l_r}) \\ \sigma_2&=&(b_{1,1}b_{1,2} \cdots b_{1,l_1}) (b_{2,1}b_{2,2} \cdots b_{2,l_2}) \cdots (b_{r,1}b_{r,2} \cdots b_{r,l_r}). \\ \end{eqnarray*} Now if $ \sigma_2 =\tau \sigma_1 \tau^{-1}$ then $ \tau= ?$

Answer 2

Hint:

If $\gamma=(a_1\,a_2\,\dots\,a_r)$ is a cycle in the decomposition of the permutation $\sigma_1$, $\tau\gamma\tau^{-1}$ is the cycle $$\bigl(\tau(a_1)\,\tau(a_2)\,\dots\,\tau(a_r)\bigr),$$ so the question is really: given any two cycles with the same length, can you find a permutation that maps one cycle onto the other?