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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 $(12)(1),(12)(2),(12)(3)$?

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    $\begingroup$ Yes. But then, you see, $(12)$ is the permutation that affects only $1$ and $2$, so it doesn't affect any other number. So $X_{(12)}$ should be immediate. $\endgroup$ – Scientifica Dec 5 '18 at 19:01
  • $\begingroup$ And so $X_{(12)}=\left \{3\right \}$? $\endgroup$ – numericalorange Dec 5 '18 at 19:17
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    $\begingroup$ That's absolutely right! $\endgroup$ – Scientifica Dec 5 '18 at 19:30
  • $\begingroup$ @Scientifica Oh, wow, I feel so happy that you helped me understand this so easily!! $\endgroup$ – numericalorange Dec 5 '18 at 19:31
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    $\begingroup$ I'm gonna post this as an answer so that your question is answered. $\endgroup$ – Scientifica Dec 5 '18 at 19:32
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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 $G_x$.

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