Let be a group which acts on the set $X$ . Then for every $x \in X$ , the stabiliser ${S t a b}_{G} (x)$ 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 $(g h) (x) = g (h (x))$ by definition of a group action, but that is $g (x) = x$ .
Associativity is inherited from the parent group.
Inverses: if $g (x) = x$ then $g^{- 1} (x) = g^{- 1} g (x) = e (x) = x$ .

Stabiliser is a subgroup