<s>[INST] Find all $\tau$ in $S_4$ such that $\tau (12)(34) \tau^{-1} = (12)(34)$ I know there should be 8 but I have only found these 6 so far, so I am missing 2:
$(1)$
$(1 2)$
$(3 4)$
$(12)(34)$
$(13)(24)$
$(14)(23)$
I know that if $\sigma$ is a $k$-cycle  and $\tau\sigma\tau^{-1} = \sigma$ then $\tau$ is a power of $\sigma$. But what can we say about $\tau$ when sigma is a product of disjoint cycles?
 [/INST] In extended notation, the permutations $\tau\in S^4$ such that 
$$\tau(1,2)(3,4)\tau^{-1} = (\tau(1),\tau(2))(\tau(3),\tau(4))=(1,2)(3,4)$$ clearly are
$$ \left(\begin{smallmatrix}1 & 2 & 3 & 4 \\ 1 & 2 & 3 & 4\end{smallmatrix}\right),\left(\begin{smallmatrix}1 & 2 & 3 & 4 \\ 1 & 2 & 4 & 3\end{smallmatrix}\right),\left(\begin{smallmatrix}1 & 2 & 3 & 4 \\ 2 & 1 & 3 & 4\end{smallmatrix}\right),\left(\begin{smallmatrix}1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3\end{smallmatrix}\right)$$
$$ \left(\begin{smallmatrix}1 & 2 & 3 & 4 \\ 3 & 4 & 1 & 2\end{smallmatrix}\right),\left(\begin{smallmatrix}1 & 2 & 3 & 4 \\ 3 & 4 & 2 & 1\end{smallmatrix}\right),\left(\begin{smallmatrix}1 & 2 & 3 & 4 \\ 4 & 3 & 1 & 2\end{smallmatrix}\right),\left(\begin{smallmatrix}1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1\end{smallmatrix}\right)$$
one just have to decide if the sets $\{1,2\}$,$\{3,4\}$ get switched or not, then if the action of the permutation on $\{1,2\}$ and $\{3,4\}$ is order-preserving or not. Three independent binary choices, hence eight chances.
The eight elements above form a non-abelian subgroup of $S_4$ with $8$ elements and exactly two elements of order $4$, thus a group isomorphic to the dihedral group $D_8$.
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