Group action

A group action is a group acting as symmetries of a set. Many deep results in group theory — Sylow's theorems, the class equation, Cayley's theorem — are corollaries of a single counting principle called the orbit-stabilizer theorem.

Definition

An action of a group G on a set X is a map G × X → X, written (g, x) ↦ g · x, satisfying

e · x = x for all x ∈ X,
(gh) · x = g · (h · x) for all g, h ∈ G, x ∈ X.

Equivalently: an action is a homomorphism G → Sym(X) to the symmetric group on X.

For x ∈ X:

The orbit of x is G·x = {g·x : g ∈ G} ⊆ X.
The stabilizer is Gₓ = {g ∈ G : g·x = x} ≤ G. A subgroup.

The action is transitive if there is a single orbit, faithful if the homomorphism G → Sym(X) is injective (equivalently, ⋂ₓ Gₓ = {e}).

Orbit-stabilizer

For a finite group G acting on X, and any x ∈ X:

|G · x|  =  [G : Gₓ]     and hence     |G|  =  |G·x| · |Gₓ|.

Proof sketch. The map gGₓ ↦ g·x is a well-defined bijection from cosets of Gₓ to the orbit of x. ∎

Burnside's lemma

If G acts on a finite set X, the number of orbits is the average number of fixed points:

|X / G|  =  (1/|G|)  Σ_{g ∈ G}  |Fix(g)|,

where Fix(g) = {x ∈ X : g·x = x}. Useful for counting colorings, necklaces, and symmetry classes.

Note on naming. This is Burnside's lemma (orbit counting). The distinct Burnside theorem (groups of order pᵃqᵇ are solvable) lives on its own page — see Burnside's theorem.

Canonical actions

Left regular action. G on itself by g · h = gh. Faithful, transitive. The embedding G ↪ Sym(G) is Cayley's theorem.
Conjugation action. G on itself by g · h = ghg⁻¹. Orbits are conjugacy classes, stabilizers are centralizers.
Coset action. G on G/H by g · (xH) = (gx)H. Transitive. Stabilizer of H is H. Kernel is the core ⋂_g gHg⁻¹, the largest normal subgroup contained in H.
Action on subgroups. G on {subgroups of G} by conjugation. Stabilizer of H is the normalizer N_G(H).

Examples

Sₙ on {1, …, n} — tautological, transitive, faithful.
Dₙ on the vertices of a regular n-gon — transitive, faithful.
GL(V) on the non-zero vectors of V — transitive iff V is a line.
G on the cosets G/H — kernel reveals the largest normal subgroup of G inside H (useful for detecting small quotients).

Applications

Class equation. Apply orbit-stabilizer to the conjugation action — see conjugacy.
Sylow theorems. All three follow from well-chosen group actions — see sylow.
Cayley's theorem. The left regular action realizes any finite group as a permutation group.