# Conjugation by a transposition for permutations

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

• @Yanko the answers in the linked post construct $\tau$ – Rylee Lyman Jun 28 at 14:42

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.
• Any element of $S_n$ can be written as a product of transpositions! That's a fairly special property. – Rylee Lyman Jun 28 at 16:16
• If we write $|\tau|$ for the minimal number of transpositions needed to write $\tau$ as a product of transpositions, since $S_n$ has $n!$ elements, there is a priori a maximum value of $|\tau|$ as $\tau$ varies. – Rylee Lyman Jun 28 at 16:18
• More interesting would be, if $\sigma$ and $\pi$ are conjugate, to find a $\tau$ with $\pi = \tau\sigma\tau^{-1}$ and $|\tau|$ the minimum possible. – Rylee Lyman Jun 28 at 16:20