Concerning the conjugation, I learned that two permutations $ \sigma,\pi\in S_n$ are conjugate if exists $\tau \in S_n $ such that: $\pi=\tau\sigma\tau^{-1}$. Also, these permutations are conjugate if and only if they have the same cycle type.
Although I know how to find the conjugate for a a permutation like in this post, I can't understand the definition of conjugation by a transposition!.
What is the difference between finding the conjugate of a permutation as in the previous link and the conjugation by a transposition? what is the role of the transposition here? how could we do this?
I would like to find a definition for the (conjugation by a transposition), In which consequence? for example, if I want to move from the permutation $ \sigma $ to $\pi$, how could I use the conjugation by a transposition.
Thanks in advance for any example or a reference
I'm not sure if "conjugate by a transposition" is a very common term, but here is what it should mean. $\pi$ and $\sigma$ in are conjugate, as you say, if there is $\tau$ such that $\pi = \tau\sigma\tau^{-1}$. If $\tau$ is a transposition (that is, $\tau = (ij)$ for some distinct $i$ and $j$ in $\{1,\dotsc,n\}$,) then $\pi$ and $\sigma$ are conjugate by a transposition.
Since if $(\sigma_1,\dotsc,\sigma_k)$ is a $k$-cycle and $\rho \in S_n$, we have $\rho(\sigma_1,\dotsc,\sigma_k)\rho^{-1} = (\sigma_{\rho(1)},\dotsc,\sigma_{\rho(k)})$, it should be simple to check whether $\sigma$ and $\pi$ are conjugate by a transposition: Write $\sigma$ and $\pi$ in cycle notation. If it is possible to arrange it so that $\sigma$ and $\pi$ differ in only two entries, then $\sigma$ and $\pi$ are conjugate by a transposition. If this is not possible, then they are not.
ETA – More generally, it should be possible to find a $\tau$ with minimal "complexity" (maybe in terms of cycle decomposition of $\tau$) with an adjustment of this method by writing $\sigma$ and $\tau$ in cycle notation with the fewest number of differing entries possible.