This topic is giving me a hard time, I was thinking an example might help clear thing up.

Lets say I was given a group $\displaystyle S_3$ which consists of $\displaystyle \{1, (1,2), (1,3), (2,3), (1,2,3), (1,3,2)\}$

and a set $\displaystyle X$ consisting of $\displaystyle \{2,3,7,9\}$

and I was asked to find the respective orbits. Then would it simply be:

$\displaystyle 1\cdot 2 = 2$ where 1 is a stabalizer

$\displaystyle (1,2)(2) = (1,2)$

$\displaystyle (1,3)(2) = (1,3)$ stabalizes 2

$\displaystyle (2,3)(2) = (2,3)$

$\displaystyle (1,2,3)(2) = (1,2,3)$

$\displaystyle (1,3,2)(2) = (1,3,2)$

so the orbit of 2 is $\displaystyle \{1,2,3\}$ with order 3. And if I do the same procedure for 3 I would get the same elements with the same order so $\displaystyle O_2 = O_3 = \{1,2,3\}$

Regarding 7 and 9, I'm thinking that all elements of $\displaystyle S_3$ stabalize 7 and 9, since they don't move, both with order 1 and $\displaystyle O_7 = \{7\}$ and $\displaystyle O_9 = \{9\}$

is this correct?

And how would $\displaystyle S_3 $ partition $\displaystyle X$?