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The stabilizer subgroup of an element x ∈ X is the subgroup of G containing exactly those elements which preserve x. We write G_{x} = {g ∈ G : gx = x}.

The orbit of a point x ∈ X is defined as the subset of X reachable from x by the actions of G. We write G(x) = {gx : g ∈ G}. Orbits may be used to form an equivalence relation on X, where two points x, y ∈ X are considered equivalent if they lie in the same orbit; that is, if y = gx for some g ∈ G.

Let H = G_{e} where e is the identity element of G.

A generic orbit G(x) has the property that it stabilizer group G_{x} is trivial, and that |G(x)| = |G|. The points in a generic orbit may thus be put into one-to-one correspondence with elements of G.