Compute all $X_{g}$ and all $G_{x}$ for $X = \left \{1, 2, 3\right \}$, $G = S_{3} = \left \{(1), (12), (13), (23), (123), (132)\right \}$.

Question

Compute all $X_{g}$ and all $G_{x}$ for $X = \left \{1, 2, 3\right \}$, $G = S_{3} = \left \{(1), (12), (13), (23), (123), (132)\right \}$.

Can someone give me a head start to this problem?

$X_{g}=\left \{x\in X: gx=x\right \}$. So how do I find $X_{(12)}$? Do I calculate $(12)(1),(12)(2),(12)(3)$?

Answer 1

Yes that's how you do it.

You can also easily see it as follows: the permutation $(12)$ affects $1$ and $2$, but no other element. So $X_{(12)}$ is immediate. The same remark allows you to quickly determine the $G_x$.