Stabiliser (of a group action)

Let the group \(G\) act on the set \(X\). Then for each element \(x \in X\), the stabiliser of \(x\) under \(G\) is \(\mathrm{Stab}_G(x) = \{ g \in G: g(x) = x \}\). That is, it is the collection of elements of \(G\) which do not move \(x\) under the action.

The stabiliser of \(x\) is a subgroup of \(G\), for any \(x \in X\). (Proof.)

A closely related notion is that of the orbit of \(x\), and the very important Orbit-Stabiliser theorem linking the two.