Conjugation in $S_n$

Question

We have that any element in $S_n$ is generated by the adjacent transpositions $(12),\dots,(n-1,n)$. I am trying to calculate the conjugation of $(ab)\in S_n$ by $\sigma\in S_n$.

So if we write sigma as a product of transpositions and $(ab)$ does not appear in this expression then we have that $\sigma(ab)\sigma^{-1}=(ab)$ I think but I am unsure as to how to proceed if this is not the case?

Cheers?

Answer 1

Besides to P..'s hint, note that if $\sigma\in S_n$ and $$\delta=(i_1,i_2,...,i_r)\in S_n$$ so $$\sigma^{-1}\delta\sigma=(i_1\sigma,...,i_r\sigma)$$

Answer 2

Let's say that $\sigma(a)=x$ and $\sigma(b)=y$. Denote $\sigma(ab)\sigma^{-1}$ by $\tau$.
Now to find $\tau$, evaluate $\tau(x),\tau(y)$ and $\tau(z) $ for $z\neq x,y$.