[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$~$.