Orbit

For a Group action of $G$ on a set $X$, and $x\in X$, the orbit of $x$ is

$$\boxed{Gx=\{\lambda (g)x:g\in G\}\subseteq X.}$$

The set of all images of $x$ under elements of $G$ - “everywhere $x$ can be moved to.”

Properties

1. Orbits partition $X$: either $Gx=Gy$ or $Gx\cap Gy=\varnothing$.
2. If $x,y$ are in the same orbit, $Gx=Gy$.
3. The action is transitive iff there is a single orbit (i.e. $Gx=X$).

(Property 1 means $X$ decomposes uniquely as $X={\mathrm{Orb}}_1\bigsqcup \cdots \bigsqcup {\mathrm{Orb}}_m$.)

Orbit Size: Orbit-Stabiliser theorem

$$|Gx|=[G:{\mathrm{Stab}}_G(x)]=|G|/|{\mathrm{Stab}}_G(x)|\text{(for finite }G\text{)}.$$

So orbit sizes always divide $|G|$.

Examples

• Regular action ($X=G$, $\lambda (g)x=gx$): single orbit $Gx=G$ - transitive.
• Conjugation action ($X=G$, $\lambda (g)x=gx{g}^{-1}$): orbits are conjugacy classes. The orbit of $e$ is $\{e\}$.
• $S_4$ on $\{1,2,3,4\}$ by $\lambda (g)i=gi$: single orbit $\{1,2,3,4\}$ - transitive.
• $D_8$ on cube corners (split top/bottom): two orbits of size 4.
• Cyclic group $⟨(123)(456)⟩$ on $\{1,2,3,4,5,6\}$: orbits $\{1,2,3\}$ and $\{4,5,6\}$.

Orbit Representatives

A set of orbit representatives is a choice of one $x_i$ from each orbit. The orbits are then $\{G{x}_1,\dots ,G{x}_m\}$, and any $G$-equivariant question can be reduced to questions about the representatives.