permotation group S13 question

Question

if a = (1 2 4 8) (3 5 7 9 11 13) (6 12) ∈ S13, and b= (1 2 3 4 5 6) (7 8 9 10) (11 12) ∈ S13 ,let σ ∈ S13 a = σbσ−1 then how much σ like this are there in S13?

Answer 1

Hint:

Prove that if $\;(i_1\;i_2\;\ldots\;i_k)\;$ is a cycle in $\;S_n\;$ and we take any element $\;\sigma\in S_n\;$ , then

$$\sigma(i_1\;i_2\;\ldots\;i_k)\sigma^{-1}=(\sigma(i_1)\;\sigma(i_2)\;\ldots\;\sigma(i_k))$$

Further hint: if $\;c_1,..,c_r\;$ are cycles (or, in fact, general permutations), then

$$\sigma(c_1\cdot\ldots\cdot c_k)\sigma^{-1}=(\sigma c_1\sigma^{-1})(\sigma c_2\sigma^{-1})\cdot\ldots\cdot(\sigma c_r\sigma c_r^{-1})$$