The Orbit Stabilizer Theorem

If is a group acting on a set the orbit of a point is the set of elements of X to which x can be moved by the elements of G. The orbit of x is denoted by The set of orbits of points under the action of form a partition of We may define an equivalence relation on the set of elements of as if there exists such that The orbits are then the equivalence classes under this relation; two elements are equivalent if and only if their orbits are the same: The set of all orbits of under the action of is written as If is a subset of we write for the set We call the subset invariant under if In that case, also operates on  is fixed if for all and all Every subset that's fixed under is also invariant under G.

For each we define the stabilizer subgroup of as the set of all elements in hat fix  This is a subgroup of though typically not a normal one. The action of on is free if and only if all stabilizers are trivial. The kernel of the homomorphism from onto the set of bijections of into is given by the intersection of the stabilizers for all Orbits and stabilizers are closely related. For a fixed consider the map from to given by The standard quotient theorem of set theory then gives a natural bijection between and Specifically, the bijection is given by This result is known as the orbit-stabilizer theorem.

Theorem The Orbit Stabilizer Theorem

Suppose that is a group acting on a set For each let be the orbit of let be the stabilizer of and let be the set of left cosets of Then for each the function defined by is a bijection. In particular, and for all Proof:
If is such that for some then and so and This shows that is well-defined.

It is clear that is surjective. If then for some and so . Thus is also injective.