# Stabilizer

(Difference between revisions)
 Revision as of 11:42, 28 February 2007 (view source) (→Example)← Older edit Revision as of 11:03, 17 May 2017 (view source)m (Style edits to align with printed edition)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''). - 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''a'' = {g ∈ G | ''a''*g = ''a''} + ''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. + 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== ==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 "stable" under the action of the stabilizer is the [[Wyckoff position]] which corresponds to that [[site symmetry|site-symmetry group]]. + 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

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.