# Stabilizer

### From Online Dictionary of Crystallography

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- | <font color="blue">Stabilisateur</font> (''Fr''). <font color="red">Stabilisator</font> (''Ge''). <font color="black">Stabilizzatore</font> (''It''). <font color="purple"> | + | <font color="blue">Stabilisateur</font> (''Fr''). <font color="red">Stabilisator</font> (''Ge''). <font color="black">Stabilizzatore</font> (''It''). <font color="purple">安定群</font> (''Ja''). <font color="brown">Стабилизатор</font> (''Ru''). <font color="green">Estabilizador</font> (''Sp''). |

## Latest revision as of 16:06, 28 November 2018

Stabilisateur (*Fr*). Stabilisator (*Ge*). Stabilizzatore (*It*). 安定群 (*Ja*). Стабилизатор (*Ru*). Estabilizador (*Sp*).

Synonym: isotropy group.

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.