a stabilizer

An orbit asks where a point can go. The stabilizer asks the opposite: which transformations leave that point exactly where it is? Pin a point on a spinning record and ask which spins keep your finger over the same groove. The stabilizer of a point is the collection of all group elements that fix that point — every move that, applied to it, brings it back to itself.

Precisely, for a group G acting on a space and a chosen point x, the stabilizer of x is the set of all g in the group with g . x = x. This set is itself a group sitting inside G — a subgroup — because the identity fixes x, the reverse of anything that fixes x also fixes x, and two moves that each fix x combine into a move that still fixes x. For the rotation group acting on the plane, the stabilizer of the centre is the whole group (every rotation fixes the centre), while the stabilizer of any other point is just the identity (only doing nothing keeps that point still). Orbit and stabilizer are tied together by a clean accounting rule, the orbit–stabilizer relationship: roughly, the bigger a point's stabilizer, the smaller its orbit, because moves spent holding the point fixed are moves not spent carrying it elsewhere.

Stabilizers measure the symmetry a geometry sees at a single point. In a homogeneous geometry — one transitive orbit — the stabilizer of a point is what is left of the group after you have used up its freedom to slide that point anywhere, and its size reflects how much 'local rotation and reflection' the geometry permits around a fixed location. This is the natural meeting point of the Erlangen program with the idea of symmetry: the full symmetry group of a figure is precisely the stabilizer of that figure under the ambient transformation group.