# Stabiliser is a subgroup

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.

• Inverses: if $$g(x) = x$$ then $$g^{-1}(x) = g^{-1} g(x) = e(x) = x$$.

Parents:

• Group action

“Groups, as men, will be known by their actions.”