[summary: Given a group $G$ acting on a set $X$, the stabiliser of some element $x \in X$ is a subgroup of $G$. ]

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

Proof

We must check the group axioms.

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