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.”