# 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.”