Let $G$ be a group which acts on the set $X$. Then for every $x\in X$, the stabiliser ${Stab}_G\left(x\right)$ is a subgroup of $G$.

Proof

We must check the group axioms.

• The identity, $e$, is in the stabiliser because $e\left(x\right)=x$; this is part of the definition of a group action.
• Closure is satisfied: if $g\left(x\right)=x$ and $h\left(x\right)=x$, then $\left(gh\right)\left(x\right)=g\left(h\right(x\left)\right)$ by definition of a group action, but that is $g\left(x\right)=x$.
• Associativity is inherited from the parent group.
• Inverses: if $g\left(x\right)=x$ then $g^{-1}\left(x\right)=g^{-1}g\left(x\right)=e\left(x\right)=x$.