Fundamental theorem of group actions

The fundamental theorem of group actions is a result relating the orbits and stabilizers of a group action.

Name

This result is also sometimes termed Burnside's theorem or the orbit-stabilizer theorem.

Statement

Statement for transitive group actions

Suppose a group $G$ acts transitively on a nonempty set $S$. Suppose $s\in S$ is a point, and $G_{s}$ denotes the isotropy subgroup of $s$ in $G$, i.e.:

$G_{s}=\{g\in G\mid g.s=s\}$.

Then, there exists a unique bijective map between the left coset space of $G_{s}$ in $G$ and the set $S$:

$\varphi :G/G_{s}\to S$

satisfying the property that it is $G$-equivariant with respect to the natural action on the left coset space; in other words, for any $g\in G$ and any left coset $hG_{s}$, we have:

$\varphi (ghG_{s})=g.\varphi (hG_{s})$.

Note that this yields:

$|G/G_{s}|=|S|$.

Combining this with Lagrange's theorem, we obtain that:

$|G|=|G_{s}||S|$.

Statement for more general group actions

Suppose $G$ is a group acting on a set $S$. Let $x\in S$, and $K$ be the orbit of $x$ under the action of $G$. Then, if $G_{x}$ denotes the stabilizer of $x$ in $G$, we have a bijection:

$G/G_{x}\to K$.

Thus:

$|G/G_{x}|=|K|$

and

$|G|=|G_{x}||K|$

Note that this follows directly from the statement about transitive group actions, because the action of $G$ restricted to the orbit of $x$ 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

• Size of conjugacy class equals index of centralizer, size of conjugacy class divides order of group
• Sum of elements in row of character table of finite group is non-negative integer

Related facts about group homomorphisms

• Normal subgroup equals kernel of homomorphism
• Fundamental theorem of group homomorphisms

Proof

Construction of the map

We first describe the map $\varphi$.

For a left coset $hG_{s}$, define:

$\varphi (hG_{s})=h.s$.

We need to prove that this is well-defined, and independent of the choice of coset representative. Thus, suppose that $h_{1},h_{2}$ are in the same left coset of $G_{s}$. Then, there exists $g\in G_{s}$ such that $h_{2}=h_{1}g$. Thus:

$h_{2}.s=(h_{1}g).s=h_{1}.(g.s)=h_{1}.s$

proving that the map is well-defined and independent of the choice of coset representative.

Proof that the map is injective

Suppose $h_{1},h_{2}\in G$ are such that $h_{1}.s=h_{2}.s$. Then, applying $h_{1}^{-1}$ to both sides yields:

$h_{1}^{-1}.(h_{1}.s)=h_{1}^{-1}.h_{2}.s\implies s=(h_{1}^{-1}h_{2}).s$

Thus, $h_{1}^{-1}h_{2}\in G_{s}$, forcing $h_{1},h_{2}$ to be in the same left coset of $G_{s}$. Thus, two elements from different left cosets cannot send $s$ to the same element.

Proof that the map is surjective

Since the action of $G$ on $S$ is transitive, every element of $S$ is expressible as $h.s$ for some $h\in G$, and hence as $\varphi (hG_{s})$.

Proof that the map is equivariant

We need to prove that:

$\varphi (ghG_{s})=g.\varphi (hG_{s})$.

The left side is $gh.s$ while the right side is $g.(h.s)$. The two are clearly equal, by the definition of a group action.