# 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.”