Fundamental theorem of group actions
- 1 Name
- 2 Statement
- 3 Related facts
- 4 Proof
This result is also sometimes termed Burnside's theorem.
Statement for transitive group actions
Suppose a group acts transitively on a nonempty set . Suppose is a point, and denotes the isotropy subgroup of in , i.e.:
Then, there exists a unique bijective map between the left coset space of in and the set :
satisfying the property that it is -equivariant with respect to the natural action on the left coset space; in other words, for any and any left coset , we have:
Note that this yields:
Combining this with Lagrange's theorem, we obtain that:
Statement for more general group actions
Suppose is a group acting on a set . Let , and be the orbit of under the action of . Then, if denotes the stabilizer of in , we have a bijection:
Note that this follows directly from the statement about transitive group actions, because the action of restricted to the orbit of is transitive.
Related facts about group actions
- Group acts on left coset space of subgroup by left multiplication
- Orbit-counting theorem
- Class equation of a group
- Class equation of a group action
Related facts about group homomorphisms
Construction of the map
We first describe the map .
For a left coset , define:
We need to prove that this is well-defined, and independent of the choice of coset representative. Thus, suppose that are in the same left coset of . Then, there exists such that . Thus:
proving that the map is well-defined and independent of the choice of coset representative.
Proof that the map is injective
Suppose are such that . Then, applying to both sides yields:
Thus, , forcing to be in the same left coset of . Thus, two elements from different left cosets cannot send to the same element.
Proof that the map is surjective
Since the action of on is transitive, every element of is expressible as for some , and hence as .
Proof that the map is equivariant
We need to prove that:
The left side is while the right side is . The two are clearly equal, by the definition of a group action.