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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!

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  • $\begingroup$ I've just had a thought, is it asking for the element $\theta=\sigma(ab)\sigma$ in which case it would be $(\sigma(a),\sigma(b))$ $\endgroup$ – Mike Davies May 22 '12 at 15:03
  • $\begingroup$ It is asking for the element $\sigma(a,b)\sigma^{-1}$. $\endgroup$ – M Turgeon May 22 '12 at 15:21
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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.

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