Find the centraliser of $(12)(34)$ in $S_4$.

Question

I am trying to find the centraliser of $(12)(34)$ in $S_4$.

I have that

$$\begin{align}C_{S_4}((12)(34)) &=\{g \in G : g(12)(34)g^{-1}=(12)(34) \} \\ &= \{g \in G : (g(1) g(2))(g(3)g(4))=(12)(34) \} \end{align}$$

Apparently the answer is $\{e, (12),(34), (12)(34),(1324),(1423),(13)(24),(14)(23) \}$ See http://www.maths.manchester.ac.uk/~rs/AlgStSol7A.pdf

I just cannot see how $(1324)$ and $(13)(24)$ can be part of the centraliser as they seem to violate $g(1) \in \{1,2\}, g(2) \in \{1,2\}$ and $g(3) \in \{3,4 \}, g(4) \in \{3,4 \}$

I was given the hint: Use the orbit-stabiliser. But I cannot see how this is any practical use

Answer 1

Hint: The centraliser of $(12)(34)$ in $S_4$ is given by what you have shown:

$$C_{S_4}((12)(34))=\{g\in G| g(12)(34)g^{-1}=(12)(34)\}$$

Firstly we note that by cycle type, conjugating it with anything will give you back a $2-2$ cycle type(since these are all of the things in the conjugacy class), so now we just need to find what doesn't shift it to one of the other elements of the conjugacy class, i.e. $(13)(24)$ and $(14)(23)$.