Difference between revisions of "Normalizer"
From Online Dictionary of Crystallography
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The normalizer of a space (or plane group) G with respect to the group ''E'' of all [[Euclidean mapping]]s (motions, isometries) in E<sup>3</sup> (or E<sup>2</sup>) is called the ''Euclidean normalizer of G'': | The normalizer of a space (or plane group) G with respect to the group ''E'' of all [[Euclidean mapping]]s (motions, isometries) in E<sup>3</sup> (or E<sup>2</sup>) is called the ''Euclidean normalizer of G'': | ||
| − | ::: N<sub> | + | ::: N<sub>E</sub>(G) := {'''S''' ∈ ''E'' | '''S'''<sup>-1</sup>G'''S''' = G} |
The Euclidean normalizers are also known as ''Cheshire groups''. | The Euclidean normalizers are also known as ''Cheshire groups''. | ||
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The normalizer of a space (or plane group) G with respect to the group ''A'' of all [[affine mapping]]s in E<sup>3</sup> (or E<sup>2</sup>) is called the ''affine normalizer of G'': | The normalizer of a space (or plane group) G with respect to the group ''A'' of all [[affine mapping]]s in E<sup>3</sup> (or E<sup>2</sup>) is called the ''affine normalizer of G'': | ||
| − | ::: N<sub> | + | ::: N<sub>A</sub>(G) := {'''S''' ∈ ''A'' | '''S'''<sup>-1</sup>GS = G} |
| + | |||
| + | == "Symmetry of the symmetry pattern" == | ||
| + | All symmetry operations of the Euclidean normalizer N<sub>E</sub>(G) map the space group onto itself. The Euclidean normaizer of a space group is therefore the group of motions that maps the pattern of symmetry elements of the space group onto itself. For this reason, it represents the ''symmetry of the symmetry pattern''. | ||
== See also == | == See also == | ||
Revision as of 17:33, 25 February 2007
Normaliseur (Fr); Normalizzatore (It).
Contents
Definition
Given a group G and one of its supergroups S, they are uniquely related to a third, intermediated group NS(G), called the normalizer of G with respect to S. NS(G) is defined as the set of all elements S ∈ S that map G onto itself by conjugation:
- NS(G) := {S ∈S | S-1GS = G}
The normalizer NS(G) may coincide wither with G or with S or it may be a proper intermediate group. In any case, G is a normal subgroup of its normalizer.
Euclidean vs. Affine normalizer
The normalizer of a space (or plane group) G with respect to the group E of all Euclidean mappings (motions, isometries) in E3 (or E2) is called the Euclidean normalizer of G:
- NE(G) := {S ∈ E | S-1GS = G}
The Euclidean normalizers are also known as Cheshire groups.
The normalizer of a space (or plane group) G with respect to the group A of all affine mappings in E3 (or E2) is called the affine normalizer of G:
- NA(G) := {S ∈ A | S-1GS = G}
"Symmetry of the symmetry pattern"
All symmetry operations of the Euclidean normalizer NE(G) map the space group onto itself. The Euclidean normaizer of a space group is therefore the group of motions that maps the pattern of symmetry elements of the space group onto itself. For this reason, it represents the symmetry of the symmetry pattern.
See also
Chapter 15 in the International Tables for Crystallography, Volume A