Group Actions Intro: Concepts, Formulas and Examples

A group action lets a group "move" the elements of a set while respecting the group structure: each g ∈ G sends each x ∈ X to g · x ∈ X, with e · x = x and (gh) · x = g · (h · x). Think of the group as allowed moves and the set as the board.

Actions turn abstract groups into concrete geometry — rotations on a square, permutations on positions. They unify counting (Burnside's lemma), structure theorems (orbit–stabilizer), and symmetry analysis in physics and chemistry under one framework.

Key Concepts

• Orbit: orb(x) = {g · x : g ∈ G}; orbits partition X.
• Stabilizer: Stab(x) = {g ∈ G : g · x = x}, the subgroup fixing x.
• Orbit–stabilizer: |orb(x)| = |G| / |Stab(x)| for finite G.
• Faithful: distinct group elements act differently. Transitive: a single orbit.
• Burnside's lemma: number of orbits = (1/|G|) · Σ_{g ∈ G} |Fix(g)|, with Fix(g) = {x : g · x = x}.

Formulas

• Orbit–stabilizer: |orb(x)| · |Stab(x)| = |G|. (Counts only; no physical units.)
• Burnside's lemma: number of orbits = (1/|G|) Σ_{g in G} |Fix(g)|.
• Limit: orbit–stabilizer needs G finite; Burnside's lemma needs G and X finite.

Worked Example

Let S_3 act on X = {1, 2, 3} naturally. For x = 1: orb(1) = {1, 2, 3} (since (12) sends 1 → 2 and (13) sends 1 → 3), so |orb(1)| = 3; Stab(1) = {e, (23)}, so |Stab(1)| = 2. Orbit–stabilizer: 3 = 6/2. Burnside check — fixed points: e fixes 3; (12), (13), (23) fix 1 each; (123), (132) fix 0. Sum = 6, orbits = 6/6 = 1: the action is transitive.

Common Misconceptions

• "The orbit of x is a subgroup." Orbits live in the set X, not in G — they are subsets of X with no algebraic structure of their own.
• "Stabilizers of different points are unrelated." They are conjugate: Stab(g · x) = g·Stab(x)·g^{-1}, so in particular they all have the same size.
• "Every action is faithful." Many useful actions are not — for example, rotations of a square act on its two diagonals with a kernel of size 2, since a half-turn fixes both diagonals as sets.

Real-World Applications

• Combinatorial counting: Burnside's lemma counts distinct necklaces, colorings, and molecules up to symmetry.
• Chemistry: molecular symmetry groups act on atoms and orbitals, predicting spectra and selection rules.
• Geometry: classification of wallpaper patterns and crystal structures is the study of group actions on the plane and space.
• Galois theory: Galois groups act on roots of polynomials; the action's transitivity detects irreducibility.

A group action makes symmetry concrete: the orbit shows where a point can travel, the stabilizer shows what holds it still, and the orbit–stabilizer equation ties the two to the size of the group.