Group action

A group action of a group $G$ on a set $X$ is a map $\lambda :G\to \mathrm{Sym}(X)$ - equivalently, a function $G\times X\to X$, $(g,x)\mapsto \lambda (g)x$ - satisfying the two axioms

Important

1. Identity: $\lambda (e)x=x$ for all $x\in X$.
2. Compatibility: $\lambda (gh)x=\lambda (g)(\lambda (h)x)$ for all $g,h\in G$ and $x\in X$.

When such a $\lambda$ exists, $X$ is called a $G$-set.

Two Equivalent Formulations

The conditions above are equivalent to: $\lambda :G\to \mathrm{Sym}(X)$ is a Homomorphism from $G$ to the symmetric group of $X$. This is the modern viewpoint - group actions = homomorphisms into permutation groups.

Standard Examples

• Regular action of $G$ on itself: $\lambda (g)x=gx$. Always transitive, stabiliser is trivial.
• Conjugation action of $G$ on itself: $\lambda (g)x=gx{g}^{-1}$. Stabiliser is the centraliser; orbits are conjugacy classes.
• Symmetric group on $\{1,\dots ,n\}$: $\lambda (g)i=g(i)$ - the natural permutation action.
• Dihedral group on the corners of an $n$-gon: rotations and reflections permute the labelled vertices.

Key Constructions

• Orbit of $x$: $Gx=\{\lambda (g)x:g\in G\}\subseteq X$.
• Stabiliser of $x$: ${\mathrm{Stab}}_G(x)=\{g\in G:\lambda (g)x=x\}\le G$.
• Orbits partition $X$.

Key Theorems

• Orbit-Stabiliser theorem: $|Gx|=|G|/|{\mathrm{Stab}}_G(x)|$ (for finite $G$).
• Cayley’s theorem: every group is a permutation group via the regular action.
• Orbit counting theorem (Burnside / Cauchy-Frobenius): $\#\text{orbits}=\frac{1}{|G|}{\sum }_g|{\mathrm{Fix}}_X(g)|$.

Transitive vs Faithful Actions

• Transitive: there’s a single orbit, $Gx=X$ for some (equivalently, every) $x$.
• Faithful: $\lambda$ is injective, i.e. only the identity fixes everything.

The kernel of an action is the set of group elements fixing every point of $X$ - the kernel of $\lambda$ as a homomorphism. A faithful action has trivial kernel.