The Orbit Stabilizer Theorem
Ifis a group acting on a setthe orbit of a pointis 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 pointsunder the action ofform a partition ofWe may define an equivalence relation on the set of elements ofasif there existssuch thatThe orbits are then the equivalence classes under this relation; two elementsare equivalent if and only if their orbits are the same:
The set of all orbits ofunder the action ofis written as
Ifis a subset ofwe writefor the setWe call the subset invariant underifIn that case,also operates onis fixed if for alland allEvery subset that's fixed under is also invariant under G.
For eachwe define the stabilizer subgroup ofas the set of all elements inhat fix
This is a subgroup ofthough typically not a normal one. The action ofonis free if and only if all stabilizers are trivial. The kernel of the homomorphism fromonto the set of bijections ofintois given by the intersection of the stabilizersfor all
Orbits and stabilizers are closely related. For a fixedconsider the map fromto given byThe standard quotient theorem of set theory then gives a natural bijection betweenandSpecifically, the bijection is given byThis result is known as the orbit-stabilizer theorem.
Theorem The Orbit Stabilizer Theorem
Suppose thatis a group acting on a setFor eachletbe the orbit oflet be the stabilizer ofand letbe the set of left cosets ofThen for each the functiondefined by is a bijection. In particular,and for all
Ifis such that for somethen and soandThis shows thatis well-defined.
It is clear thatis surjective. Ifthenfor someand so . Thusis also injective.