# Stabilizer

### From Online Dictionary of Crystallography

Stabilisateur (*Fr*); Stabilisator (*Ge*); Stabilizzatore (*It*); 安定部分群 (*Ja*).

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.