Fundamental theorem of group actions
Contents
Name
This result is also sometimes termed Burnside's theorem.
Statement
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:
.
Thus:
and
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
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
Applications
Related facts about group homomorphisms
Proof
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.