# 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''). |

− | 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 | + | 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<sub>''a''</sub> = {g ∈ G | ''a''*g = ''a''} | + | ''G''<sub>''a''</sub> = {''g'' ∈ ''G'' | ''a''*g = ''a''} |

− | is called the '''stabilizer''' of A. G<sub>''a''</sub> is the set of all elements of G which leave ''a'' unchanged or 'stable'. G<sub>''a''</sub> is a [[subgroup]] of G. | + | is called the '''stabilizer''' of ''A''. ''G''<sub>''a''</sub> is the set of all elements of ''G'' which leave ''a'' unchanged or 'stable'. ''G''<sub>''a''</sub> is a [[subgroup]] of ''G''. |

==Example== | ==Example== | ||

− | The [[site symmetry|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|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 | + | The [[site symmetry|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|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|site-symmetry group]]. |

==See also== | ==See also== |

## Revision as of 11:03, 17 May 2017

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.