Stabilizer

From Online Dictionary of Crystallography

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<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'').
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<font color="blue">Stabilisateur</font> (''Fr''). <font color="red">Stabilisator</font> (''Ge''). <font color="black">Stabilizzatore</font> (''It''). <font color="purple">安定群</font> (''Ja''). <font color="brown">Стабилизатор</font> (''Ru''). <font color="green">Estabilizador</font> (''Sp'').
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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:
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Synonym: isotropy group.
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G<sub>''a''</sub> = {g &isin; G | ''a''*g = ''a''}
 
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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.
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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
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''G''<sub>''a''</sub> = {''g'' &isin; ''G'' | ''a''*g = ''a''}
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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==
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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]].
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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==

Latest revision as of 16:06, 28 November 2018

Stabilisateur (Fr). Stabilisator (Ge). Stabilizzatore (It). 安定群 (Ja). Стабилизатор (Ru). Estabilizador (Sp).


Synonym: isotropy group.


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.

See also