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For $i \in \{1,2,...,n-1\}$ let $\tau_i \in S_n$ be the transposition of $i$ and $i+1$. I want to know wether $$\tau_i \circ ... \circ \tau_2 \circ \tau_1 \circ \tau_2 \circ ... \circ \tau_i $$ is a permutation that is also a Transposition.

I guess I know how to solve it now, thanks! There is already a nice idea below.

calculatormathematical
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    Did you try it for i=1,2,3? – Isomorphism May 19 '21 at 12:25
  • Have you tried induction? Is $\tau_1$ a transposition? Is $\tau_2\circ \tau_1\circ \tau_2$ a transposition? Which one specifically? Now, try to form the correct induction hypothesis and induction argument to talk about $\tau_1\circ\dots\circ\tau_2\circ\tau_1\circ\dots\circ\tau_i$ – JMoravitz May 19 '21 at 12:26
  • The expression $\tau_3(k)$ makes sense for all $k\in{1,\dots,n}$. You have $\tau_3(k)=k$ for $k\notin{3,4}$, $\tau_3(3)=4$ and $\tau_3(4)=3$. In particular $\tau_3(2)=2$. – Christoph May 19 '21 at 12:30
  • Hello, $\tau_3(2)$ would just be $2$. In general these transpisitions fix all of the elements except for the two that they are transposing. Perhaps this can help you solve the problem by hand! I added a more "theoretical" solution, but I also recommend doing it by hand ! – Asinomás May 19 '21 at 12:30
  • "and now I can only plug in 3 can't I? Because $\tau_3(2)$ makes no sense to me." Why does it make no sense to you? Recall that a transposition $\tau_3$ can be represented by two-line notation $\begin{pmatrix}1&2&3&4&\dots\1&2&4&3&\dots\end{pmatrix}$ Alternatively phrased, a transposition $(3~4)$ can also be written with all the invisible "one-cycles" as $\color{grey}{(1)(2)}(3~4)\color{grey}{(5)\cdots}$. You have that $\tau_3(2)=2$... that it doesn't change it – JMoravitz May 19 '21 at 12:31

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This is a particular case of conjugating a permutation.

You have a permutation $\sigma$ and another permutation $\rho$ and you are considering the permutation $\rho \sigma \rho^{-1}$.

In this case we can let $\rho =\tau_i \dots \tau_2$ and $\sigma=\tau_1$.

There is a general result that tells us that given a permutation $\sigma$ the set of permutations that are of the form $\rho \sigma \rho^{-1}$ are precisely those that have the same cycle type as $\sigma$. So in this case it would be all of the transpositions.

Asinomás
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