a group action

A group is an abstract gadget of 'moves' that can be combined and undone. A group action is the bridge that lets such a group actually do something to a concrete space — it tells you how each group element moves the points around. When the group of rotations acts on the plane, each rotation is told exactly which point goes where; that telling-where is the action.

Precisely, an action of a group G on a set X is a rule that assigns to each group element g and each point x a new point, written g . x, in a way that respects the group's structure. Two demands make it 'respect the structure'. First, the identity element e of the group must do nothing: e . x = x for every point x. Second, combining must agree with composing: if you act by h and then by g, you get the same result as acting once by the combined element g times h, that is (gh) . x = g . (h . x). These two rules are exactly what guarantee that the abstract group laws translate faithfully into honest moves of the space. In the Erlangen picture, the transformation group acts on the geometric space, and everything else — orbits, stabilizers, invariants — is built from this single notion.

It is worth separating the group itself from its action. The same abstract group can act on many different spaces, and sometimes a nontrivial group acts so weakly that some elements still move nothing — that is a sign the action is not faithful. The geometry you get depends not just on which group you choose but on how it acts. The Erlangen program quietly assumes a fixed, faithful, transitive action on a homogeneous space: faithful so distinct transformations really differ, transitive so the space looks the same from every point.