Conjugates of Transpositions 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!

You were close in your comment (it was possibly a typo). 
Let $G$ be a group and let $g, h \in G$. The conjugate of $g$ by $h$ is the element $hgh^{-1}$. 
In this case, the group is $S_n$, $g = (a\ b)$ and $h = \sigma$. Therefore, the conjugate of $(a\ b)$ by $\sigma$ is $\sigma(a\ b)\sigma^{-1}$ which is, as you noted, $(\sigma(a)\ \sigma(b))$. 
In general, given $\sigma \in S_n$ and a cycle $c = (a_1\ \dots\ a_k) \in S_n$, the conjugate of $c$ by $\sigma$ is $(\sigma(a_1)\ \dots\ \sigma(a_k))$; see here for example.
