Group Theory/Cardinality identities for finite representations

Definition (permutation representation):

Let be a group and let be a set. A permutation representation of on is a representation , where the automorphisms of are taken in the category of sets (that is, they are just bijections from to itself).

Definition (pointwise stabilizer):

Let be a group, let be an algebraic variety and let be an instance of . Suppose that is a representation in the category defined by . Let . Then the pointwise stabilizer of is given by


Proposition (transitive permutation representation is equivalent to right multiplication on quotient by stabilizer):

Let be a group, let be a set and suppose that we have a permutation representation which is transitive. Let be arbitrary and let be the pointwise stabilizer of . Consider the action by left multiplication, where is the set of left cosets of (which is in fact never a normal subgroup in this situation, unless the action is trivial, because ). Then there exists a -isomorphism from to .

Proof: We define as follows: shall be mapped to . First, we show that this map is well-defined. Indeed, suppose that we take . Then is mapped to . Then we note that the map is surjective by transitivity. Finally, it is also injective, because whenever , we have by applying to both sides and using a property of a group action, and thus , that is to say . That follows immediately from the definition, so that we do have an isomorphism of representations.

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

Theorem (orbit-stabilizer theorem):

Let be a group, and let be a permutation representation on a set . Then


Proof: acts transitively on . The above -isomorphism between and is bijective as an isomorphism in the category of sets. But the notation stood for .

Theorem (class equation):

Let be a finite group and let be a permutation representation on the finite set . Then


where are the orbits of , and for . (We also say that are a system of representatives for the orbits of .)

Proof: acts transitively on each orbits, and the orbits partition . Hence, by the orbit-stabilizer theorem,


Definition (fixed point set):

Let be a group that acts on a set , and let be a subset of . Then the fixed point set of is defined to be