As part of an assignment, I need to calculate a bunch of stabilisers. Could you please just tell me if I'm doing one of them right?
Let $G$ be a group with elements $(12)(34), (13)(24), (14)(23),e$. $X$ is a set with lots of elements, but I guess one will suffice for me to see if I'm doing this right, $(1,2) \in X$.
I then calculate $(1,2)(3,4) \circ (1,2) = (3,4)$, $(1,3)(2,4) \circ (1,2)= (1,4,2,3)$, $(1,4)(2,3) \circ (1,2)=(1,3,2,4)$, $e \circ (1,2) = (1,2)$. Hence, $G_{(1,2)}=\{e\}$.
Is the above correct? Thanks!
It looks right. Notice that the index of the stabilizer is indeed $4$, which is the order of the orbit.