MTH3003 Lecture 15

Last lecture we proved Cayley’s Theorem - every group is a permutation group via its regular action. Today we develop the two key invariants of a Group action: the orbit (where a point can travel) and the stabiliser (which group elements pin it). The two are connected by the Orbit-Stabiliser Theorem, which converts size questions about orbits into Lagrange-style index calculations on $G$.

Orbits and Stabilisers

Orbit

Let $X$ be a $G$-set. The orbit of $x\in X$ is

$$Gx=\{\lambda (g)x:g\in G\},$$

the set of all images of $x$ under the action of $G$. If $Gx=X$ for some (equivalently, every) $x\in X$, the action is transitive.

Stabiliser

Fix $x\in X$. The stabiliser of $x$ is

$${\mathrm{Stab}}_G(x)=\{g\in G:\lambda (g)x=x\},$$

the set of group elements that fix $x$.

Two complementary objects: the orbit lives in $X$ (where $x$ goes), the stabiliser lives in $G$ (which group elements pin $x$).

Worked Examples

Example: $S_4$ Acting on $\{1,2,3,4\}$

Take $G=S_4$, $X=\{1,2,3,4\}$, action $\lambda (g)i=gi$ (the natural permutation action).

• Orbits. $G\cdot 1=\{1,2,3,4\}=G\cdot 2=G\cdot 3=G\cdot 4$. So there is one orbit - the action is transitive.
• Stabilisers. ${\mathrm{Stab}}_G(1)=\{e,(234),(243),(23),(24),(34)\}$ - the six permutations of $\{2,3,4\}$. So ${\mathrm{Stab}}_G(1)\cong S_3$. Similarly ${\mathrm{Stab}}_G(i)\cong S_3$ for any $i$.

Example: $D_8$ Acting on a Cube

The natural action of $D_8$ on a cube splits the eight corners into two orbits of size four each - the “top” four corners and the “bottom” four. The action is not transitive.

For corner $1$: ${\mathrm{Stab}}_{D_8}(1)=\{e,\sigma {\rho }^3\}$ - only two symmetries fix corner $1$.

Example: Regular Action

Take $X=G$ and $\lambda (g)x=gx$ (left multiplication). For any $x\in G$,

• Orbit. $Gx=\{gx:g\in G\}=G$ - the regular action is always transitive.
• Stabiliser. $gx=x\iff g=e$, so ${\mathrm{Stab}}_G(x)=⟨e⟩$.

This is what makes Cayley’s Theorem work: the kernel of the regular action is trivial.

Example: Conjugation Action

Take $X=G$ and $\lambda (g)x=gx{g}^{-1}$.

• Orbit of $e$. $G\cdot e=\{ge{g}^{-1}:g\in G\}=\{e\}$ - the identity is alone in its orbit, so the conjugation action is not transitive (unless $G$ is trivial).
• Stabiliser of $x$. ${\mathrm{Stab}}_G(x)=\{g:gx{g}^{-1}=x\}=\{g:gx=xg\}$ - the elements that commute with $x$. This special subgroup is called the centraliser of $x$ in $G$:

$$C_G(x)=\{g\in G:gx=xg\}.$$

Abelian groups under conjugation

If $G$ is abelian, every element commutes, so $gx{g}^{-1}=x$ for all $g,x$. Hence each conjugacy orbit is just $\{x\}$, and every element is its own centraliser: $C_G(x)=G$.

Properties of Orbits and Stabilisers

Proposition

Let $X$ be a $G$-set and $x,y\in X$.

1. Either $Gx=Gy$ or $Gx\cap Gy=\varnothing$ - orbits are either equal or disjoint.
2. If $x$ and $y$ lie in the same orbit, then $Gx=Gy$.
3. If $Y$ is an orbit and $x\in Y$, then $Gx=Y$.
4. ${\mathrm{Stab}}_G(x)\le G$ - the stabiliser is a subgroup.

Property 1 means the orbits partition $X$:

$$X={\mathrm{Orb}}_1\bigsqcup {\mathrm{Orb}}_2\bigsqcup \cdots \bigsqcup {\mathrm{Orb}}_n.$$

Picking one representative $x_i$ from each ${\mathrm{Orb}}_i$ gives a set of orbit representatives. Once you know the partition into orbits, you’ve decoded the global geometry of the action.

The Orbit-Stabiliser Theorem

The headline result, connecting orbit size to stabiliser index.

Orbit-Stabiliser Theorem

Let $G$ be a group and $X$ a $G$-set. For all $x\in X$,

$$\boxed{|Gx|=[G:{\mathrm{Stab}}_G(x)].}$$

If $G$ is finite, $|Gx|=|G|/|{\mathrm{Stab}}_G(x)|$.

Proof

Set $S={\mathrm{Stab}}_G(x)$. Define a map from cosets of $S$ to elements of $Gx$ by $gS\mapsto \lambda (g)x$. We show this is a well-defined bijection.

For $g,h\in G$:

$$\lambda (g)x=\lambda (h)x⟺\lambda (h^{-1}g)x=x⟺h^{-1}g\in S⟺gS=hS.$$

(Used: $\lambda$ is a homomorphism, the equivalence in the last step is the standard coset characterisation from earlier in the course.)

So the number of distinct elements of $Gx$ equals the number of distinct left cosets of $S$ in $G$ - that’s $[G:S]$ by definition. For finite $G$, Lagrange’s theorem gives $[G:S]=|G|/|S|$. $■$

Why This Is Powerful

The formula reduces orbit-counting to a divisibility check. Common applications:

• Compute orbit sizes when you know the stabiliser.
• Show certain transitive actions don’t exist via a divisibility obstruction.

Worked Examples of Orbit-Stabiliser

Example: $S_4$ on $\{1,2,3,4\}$ Revisited

$G=S_4$, $|G|=24$. $|G\cdot 1|=4$ (transitive). The theorem requires $|G|/|{\mathrm{Stab}}_G(1)|=4$, i.e. $|{\mathrm{Stab}}_G(1)|=24/4=6$. We computed ${\mathrm{Stab}}_G(1)\cong S_3$ has six elements. ✓

Example: No Transitive Action of a Group of Order 20 on a Set of Size 6

Suppose $|G|=20$ and $G$ acts transitively on a set $X$ with $|X|=6$. Then $|Gx|=6$ for the chosen $x$, and Orbit-Stabiliser gives

$$6=\frac{20}{|{\mathrm{Stab}}_G(x)|},$$

forcing $|{\mathrm{Stab}}_G(x)|=20/6$, which is not an integer. Contradiction. So no such transitive action exists.

Example: Can $D_{10}$ Act Transitively on a Triangle?

Suppose $D_{10}$ (order $10$) acts transitively on $X=\{\text{three corners}\}$. Then $|Gx|=3$, so $|{\mathrm{Stab}}_{D_{10}}(x)|=10/3\notin \mathbb{Z}$. No.

Pattern

Any transitive action of $G$ on $X$ requires $|X|||G|$. So if $|X|\nmid |G|$, no transitive action exists. Useful for ruling out symmetry types instantly.

Example: $S_n$ Acting on Pairs

Let $G=S_n$ act on $Y\times Y$ where $Y=\{1,\dots ,n\}$, via $\lambda (g)(x,y)=(gx,gy)$.

What is $|G\cdot (1,1)|$? The stabiliser of $(1,1)$ consists of permutations fixing both coordinates, i.e. fixing $1$ - that’s $\mathrm{Sym}(\{2,\dots ,n\})$, with $|.|=(n-1)!$. So

$$|G\cdot (1,1)|=\frac{n!}{(n-1)!}=n.$$

Indeed the orbit of $(1,1)$ is $\{(i,i):i\in Y\}$ - the diagonal, which has $n$ elements.

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Pre-Lecture Notes from mth3003 lecture notes 15.pdf

• Orbit $Gx=\{\lambda (g)x:g\in G\}\subseteq X$. Transitive action: $Gx=X$.
• Stabiliser ${\mathrm{Stab}}_G(x)=\{g\in G:\lambda (g)x=x\}\le G$ - it’s a subgroup of $G$.
• Orbits partition $X$: either $Gx=Gy$ or $Gx\cap Gy=\varnothing$.
• The conjugation-action stabiliser ${\mathrm{Stab}}_G(x)=C_G(x)$ is the centraliser - elements commuting with $x$.
• Orbit-Stabiliser theorem: $|Gx|=[G:{\mathrm{Stab}}_G(x)]=|G|/|{\mathrm{Stab}}_G(x)|$ (for finite $G$).
• Proof technique: bijection between cosets $gS$ of $S={\mathrm{Stab}}_G(x)$ and elements $\lambda (g)x$ of the orbit.
• Application: rule out transitive actions when $|X|\nmid |G|$.
• Next lecture: Burnside-type orbit-counting and applications.