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
Ga = {g ∈ G | 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.