Let $\tau$ be a transposition (2-cycle) in $S_n$. Prove that for all permutations $\sigma$ in $S_n$, that $\sigma\tau\sigma^{-1}$ is a transposition.

Question

Let $\tau$ be a transposition (2-cycle) in $S_n$. Prove that for all permutations $\sigma$ in $S_n$, that $\sigma\tau\sigma^{-1}$ is a transposition.

It is clear to me that the order is 2, but it is not necessarily clear that $\tau$ is a transposition.

Answer 1

Two permutations are conjugates of each other iff they have the same cycle structure.

Actually you can easily prove that if $\tau = (a_1,a_2,...,a_k)$ then $\sigma\tau\sigma^{-1} = (\sigma(a_1),\sigma(a_2),...,\sigma(a_k))$

Answer 2

If $\tau$ transposes $i$ and $j$, leaving all other values unchanged, then $\sigma\tau\sigma^{-1}$ transposes $\sigma(i)$ and $\sigma(j)$ and leaves all other values unchanged. This follows from the definition of composition of permutations: first do $\sigma^{-1}$, then do $\tau$, then to $\sigma$.