Stabilizer

Stabilisateur (Fr); Stabilisator (Ge); Stabilizzatore (It); 安定部分群 (Ja); Стабилизатор (Ru); Estabilizador (Sp).

Let G be a group which acts on a set A by a composition law *, and let a be a given element of A. Then the set

G_a = {g ∈ G | a*g = a}

is called the stabilizer of A. G_a is the set of all elements of G which leave a unchanged or 'stable'. G_a is a subgroup of G.

Example

The site-symmetry group of a Wyckoff position is the stabilizer of that position. In this example, G is the space group, the stabilizer is the site-symmetry group, the set A is the set of triples of x,y,z coordinates (set of points in the three-dimensional space), the element a that is 'stable' under the action of the stabilizer is the Wyckoff position which corresponds to that site-symmetry group.

See also

• Centralizer
• Normalizer