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Chemistry LibreTexts

Stabilizer

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.