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

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