Group Actions: Orbits, Stabilisers, and Counting

A group is born to act: to permute a set while preserving its structure. Group actions turn abstract group elements into concrete symmetries and yield some of the most useful counting tools in finite group theory — the orbit–stabiliser theorem, Cayley's theorem, the class equation, and Burnside's lemma. This page builds on homomorphisms.md; the linear special case (actions on vector spaces) opens reps-basics.md.

Actions

A (left) group action of $G$ on a set $X$ is a map $G\times X\to X$, $(g,x)\mapsto g\cdot x$, satisfying $$e\cdot x=x,(g_1{g}_2)\cdot x=g_1\cdot (g_2\cdot x).$$ Equivalently — and more revealingly — an action is a homomorphism $$\rho :G\to \mathrm{Sym}(X)$$ into the symmetric group of $X$: each $g$ acts as a permutation $\rho (g)$, and the action axioms are exactly the homomorphism axioms. The action is faithful if $\rho$ is injective (distinct elements act differently), i.e. the kernel $\{g:g\cdot x=x\forall x\}$ is trivial.

Orbits and stabilisers

For $x\in X$:

• the orbit ${\mathcal{O}}_x=\{g\cdot x:g\in G\}\subseteq X$ is everything $x$ can be moved to;
• the stabiliser $G_x=\{g\in G:g\cdot x=x\}\le G$ is the subgroup fixing $x$.

Orbits partition $X$ (being an equivalence relation $x\sim y$ iff $y=g\cdot x$). The action is transitive if there is a single orbit ($X={\mathcal{O}}_x$).

Orbit–stabiliser theorem. For any $x$, the map $g{G}_x\mapsto g\cdot x$ is a bijection between the cosets of the stabiliser and the orbit: $$|{\mathcal{O}}_x|=[G:G_x]=\frac{|G|}{|G_x|}(\text{finite }G).$$

Thus every orbit size divides $|G|$. This one identity powers almost every counting argument below and in sylow.md.

Cayley's theorem

Let $G$ act on itself by left multiplication, $g\cdot x=gx$. This action is faithful (if $gx=x$ for all $x$ then $g=e$), so the associated homomorphism $G\to \mathrm{Sym}(G)$ is injective:

Cayley's theorem. Every group embeds in a symmetric group; a finite group of order $n$ embeds in $S_n$.

So abstract groups are "no more general" than permutation groups — though the embedding into $S_n$ is usually far from the most economical realisation.

Conjugation and the class equation

Let $G$ act on itself by conjugation, $g\cdot x=gx{g}^{-1}$. Then:

• orbits are the conjugacy classes $\mathrm{Cl}(x)=\{gx{g}^{-1}:g\in G\}$;
• the stabiliser of $x$ is its centraliser $C_G(x)=\{g:gx=xg\}$;
• the fixed points are exactly the centre $Z(G)$ (elements in a class by themselves).

Orbit–stabiliser gives $|\mathrm{Cl}(x)|=[G:C_G(x)]$, and summing over the distinct classes yields the

Class equation. $$|G|=|Z(G)|+\sum _i[G:C_G(x_i)],$$ where the $x_i$ are representatives of the conjugacy classes of size $>1$.

A signature application. If $|G|=p^n$ is a prime power ($n\ge 1$), every term $[G:C_G(x_i)]$ in the sum is a positive power of $p$, so $p$ divides $|Z(G)|$; hence a non-trivial $p$-group has non-trivial centre, $Z(G)\ne \{e\}$. This fact seeds the Sylow theory and the nilpotency of $p$-groups in series-solvable.md.

Burnside's counting lemma

To count orbits — e.g. distinct colourings up to symmetry — average the number of fixed points:

Burnside's lemma. For a finite group $G$ acting on a finite set $X$, the number of orbits is $$\#\{\text{orbits}\}=\frac{1}{|G|}\sum _{g\in G}|\mathrm{Fix}(g)|,>\mathrm{Fix}(g)=\{x:g\cdot x=x\}.$$

Proof sketch. Count pairs $\{(g,x):g\cdot x=x\}$ two ways — by $g$ (giving ${\sum }_g|\mathrm{Fix}(g)|$) and by $x$ (giving ${\sum }_x|G_x|=|G|\cdot \#\text{orbits}$ via orbit–stabiliser). $\square$

Worked example. Colour the $4$ vertices of a square with $2$ colours, counting rotations (the group ${\mathbb{Z}}_4$) as equivalent. The fixed-colouring counts are $|\mathrm{Fix}(e)|=2^4=16$, $|\mathrm{Fix}(r)|=|\mathrm{Fix}(r^3)|=2$ (all-same "necklace" forced), $|\mathrm{Fix}(r^2)|=2^2=4$. Burnside gives $$\frac{1}{4}(16+2+4+2)=6$$ distinct colourings. (Extending the group to the full dihedral $D_4$ that also allows reflections gives the classic answer $6$ for necklaces — the reflection terms recount to the same value here.) Refining Burnside to track colour multiplicities gives Pólya enumeration.

Where actions reappear

• Linear actions — an action on a vector space that respects addition and scaling is a representation; that is the whole of reps-basics.md and character-theory.md.
• Conjugation actions drive the Sylow theorems.
• Smooth actions of Lie groups on manifolds are the setting for homogeneous spaces and the physics of gauge symmetry.

References

• Dummit & Foote, Abstract Algebra, §1.7, §4.1–4.3.
• Rotman, An Introduction to the Theory of Groups, Ch. 3.