# Stabilizer

### From Online Dictionary of Crystallography

##### Revision as of 11:03, 17 May 2017 by BrianMcMahon (talk | contribs) (Style edits to align with printed edition)

Revision as of 11:03, 17 May 2017 by BrianMcMahon (talk | contribs) (Style edits to align with printed edition)

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.