Definition
Let be a group that acts on a set , and let .
Then is called invariant if
Relation to orbits
For a group that acts on a set , a subset is -invariant if and only if U is the union of orbits of the action of , i.e.
Proof
Let be a union of orbits, and let .
Then for some and .
Note
so is -invariant.
Next, let be -invariant.
For every , since .
Then by invariance, , so