Stabiliser

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

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

The set of group elements that fix the point $x$.

Properties

• ${\mathrm{Stab}}_G(x)\le G$ - always a subgroup.
• In a transitive action, all stabilisers have the same size: $|{\mathrm{Stab}}_G(x)|=|{\mathrm{Stab}}_G(y)|$ for all $x,y$.
• For points in the same orbit, the stabilisers are conjugate: if $y=\lambda (g)x$, then ${\mathrm{Stab}}_G(y)=g{\mathrm{Stab}}_G(x)g^{-1}$.

Orbit-Stabiliser theorem

$$|Gx|\cdot |{\mathrm{Stab}}_G(x)|=|G|\text{(for finite }G\text{)}.$$

So stabiliser size and orbit size are inversely related.

Examples

• Regular action of $G$ on itself: ${\mathrm{Stab}}_G(x)=\{e\}$ for all $x$ (trivial).
• Conjugation action of $G$ on itself: ${\mathrm{Stab}}_G(x)=C_G(x)$, the centraliser of $x$ - elements that commute with $x$.
• $S_4$ on $\{1,2,3,4\}$, action on point $1$: ${\mathrm{Stab}}_{S_4}(1)=\mathrm{Sym}(\{2,3,4\})\cong S_3$, size $6$. Orbit-Stabiliser: $|S_4\cdot 1|=24/6=4=|\{1,2,3,4\}|$. ✓
• $D_{2n}$ on corners of an $n$-gon, action on corner $1$: stabiliser is $\{e,r\}$ where $r$ is the reflection through corner $1$.

Distinction From Fixed-Point Set

Don’t confuse:

• ${\mathrm{Stab}}_G(x)\subseteq G$ - group elements fixing one specific point $x$.
• ${\mathrm{Fix}}_X(g)\subseteq X$ - points fixed by one specific group element $g$.

Different ambient sets, different roles. Both appear in the Orbit counting theorem.