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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

Noah16
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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.

Rylee Lyman
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  • Great clarification. I am wondering why we use them? what makes them special (is there any property)? – Noah16 Jun 28 '19 at 15:53
  • Also, (I think this is much harder) can we bound the maximum number of transpositions used for conjugation between two permutations? – Noah16 Jun 28 '19 at 15:56
  • Any element of $S_n$ can be written as a product of transpositions! That's a fairly special property. – Rylee Lyman Jun 28 '19 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 '19 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 '19 at 16:20
  • Excuse me, I think that I didn't understand your point! It is known that any permutation could be written as a product of transpositions but my question is why it is more special to conjugate by a transposition than to conjugate by any other permutation? What more benefits I can get from the conjugation by a transposition? Thanks a lot for your cooperation – Noah16 Jun 28 '19 at 16:39
  • I don't know of any reason to think it's more special. – Rylee Lyman Jun 28 '19 at 16:46
  • Yous wrote: "If we write |τ| for the minimal number of transpositions needed to write τ as a product of transposition"... but τ is already a transposition? isn't it? Can I write it as a product of transpositions or you mean as a product of simple transpositions? – Noah16 Jun 28 '19 at 17:33
  • I meant $\tau$ to be an arbitrary element of $S_n$ when I wrote that bit. – Rylee Lyman Jun 28 '19 at 20:46