# Difference between revisions of "Stabilizer"

### From Online Dictionary of Crystallography

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

## Revision as of 12:42, 13 October 2017

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.