# Math Help - simple question regarding orbits

1. ## simple question regarding orbits

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 $S_3$ which consists of $\{1, (1,2), (1,3), (2,3), (1,2,3), (1,3,2)\}$

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

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

$1\cdot 2 = 2$ where 1 is a stabalizer
$(1,2)(2) = (1,2)$
$(1,3)(2) = (1,3)$ stabalizes 2
$(2,3)(2) = (2,3)$
$(1,2,3)(2) = (1,2,3)$
$(1,3,2)(2) = (1,3,2)$

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

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

is this correct?

And how would $S_3$ partition $X$?

2. consider the action of S_3 on X.