Group Theory/Cardinality identities for finite representations

We are now in a position to derive some standard formulae for permutation representations.

{{definition|fixed point set|Let $$G$$ be a group that acts on a set $$X$$, and let $$S \subseteq G$$ be a subset of $$G$$. Then the fixed point set of $$S$$ is defined to be
 * $$\operatorname{Fix}(S) := \{x \in X| \forall s \in S: sx = x$$.}}