Groupprops, The Group Properties Wiki (pre-alpha)

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Contents 1 Name 2 Statement 2.1 Statement for transitive group actions 2.2 Statement for more general group actions 3 Related facts 3.1 Related facts about group actions 3.2 Applications 3.3 Related facts about group homomorphisms 4 Proof 4.1 Construction of the map 4.2 Proof that the map is injective 4.3 Proof that the map is surjective 4.4 Proof that the map is equivariant

Name

This result is also sometimes termed Burnside's theorem.

Statement

Statement for transitive group actions

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

.

Then, there exists a unique bijective map between the left coset space of G_s in G and the set 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 and any left coset hG_s, we have:

.

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 , 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:

.

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

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 .

For a left coset hG_s, define:

.

We need to prove that this is well-defined, and independent of the choice of coset representative. Thus, suppose that h₁,h₂ are in the same left coset of G_s. Then, there exists such that h₂ = h₁g. Thus:

h₂.s = (h₁g).s = h₁.(g.s) = h₁.s

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 h₁.s = h₂.s. Then, applying to both sides yields:

Thus, , forcing h₁,h₂ 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 , and hence as .

Proof that the map is equivariant

We need to prove that:

.

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.