vid.

If Group acts on a set $S$, it will also act on the set of all subsets (Power set).

Definition. Basically $g\cdot U = \{ g\cdot u | u \in U\}$ (the set obtained by aplying $g$ to the elements of the subset). This defines a Group action

Can also restrict action to the set of subsets of a given cardinality. This is because the group action is always an injection (as can be seen by thinking of inverses), so it preserves set cardinality.

We can look at the Stabilizer subgroup, $G_U$

Proposition. Let $G$ acnt on $S$. Let $U \subset S$. Then $G= G_U$ $\Leftrightarrow$ $U$ is an union of some $G$ orbits on $S$.