# Stabilizer

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 Revision as of 11:03, 17 May 2017 (view source)m (Style edits to align with printed edition)← Older edit Revision as of 12:42, 13 October 2017 (view source) (languages)Newer edit → Line 1: Line 1: - Stabilisateur (''Fr''). Stabilisator (''Ge''). Stabilizzatore (''It''). 安定部分群 (''Ja''). + Stabilisateur (''Fr''); Stabilisator (''Ge''); Stabilizzatore (''It''); 安定部分群 (''Ja''); Стабилизатор (''Ru''); Estabilizador (''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

Ga = {gG | a*g = a}

is called the stabilizer of A. Ga is the set of all elements of G which leave a unchanged or 'stable'. Ga 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.