I've been asked the following.
Assume $S_n$ is generated by the adjacent transpositions $(1,2),(2,3),...,(n-1,n)$
Let $\sigma \in S_n$. Calculate the conjugate of the transposition $(a,b)$ by $\sigma$.
Now I know there are $\frac{n(n-1)}{2}$ transpositions in $S_n$ so I'd imagine these are the conjugates of $(a,b)$. The thing throwing me is that it just asks for the conjugate i.e. singular so am I wrong and it is actually just a single element they are looking for? As a single element $(1,b)$ is I guess as $(1,a)(a,b)(1,a)=(1,b)$.
Thanks in advance!