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

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

• 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. – Scientifica Dec 5 '18 at 19:01
• And so $X_{(12)}=\left \{3\right \}$? – numericalorange Dec 5 '18 at 19:17
• That's absolutely right! – Scientifica Dec 5 '18 at 19:30
• @Scientifica Oh, wow, I feel so happy that you helped me understand this so easily!! – numericalorange Dec 5 '18 at 19:31
• I'm gonna post this as an answer so that your question is answered. – Scientifica Dec 5 '18 at 19:32

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 $$G_x$$.