Stabilizer

From Online Dictionary of Crystallography

(Difference between revisions)
Jump to: navigation, search
(Example)
Line 10: Line 10:
==Example==
==Example==
-
The [[site symmetry|site-symmetry group]] of a [[Wyckoff position]] is the stabilizer of that position.  
+
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 11:42, 28 February 2007

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.

See also