Difference between revisions of "Normalizer"
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− | <font color="blue"> Normaliseur </font> (''Fr'') | + | <font color="blue">Normaliseur</font> (''Fr''). <font color="red">Normalisator</font> (''Ge''). <font color="black">Normalizzatore</font> (''It''). <font color="purple">正規化群</font> (''Ja''). <font color="green">Normalizador</font> (''Sp''). |
== Definition == | == Definition == | ||
− | Given a group G and one of its supergroups S, they are uniquely related to a third, intermediated group N<sub>S</sub>(G), called the '''normalizer of G with respect to S'''. N<sub>S</sub>(G) is defined as the set of all elements '''S''' ∈ S that map G onto itself by conjugation: | + | Given a group ''G'' and one of its supergroups ''S'', they are uniquely related to a third, intermediated group ''N''<sub>''S''</sub>(''G''), called the '''normalizer of ''G'' with respect to ''S'''''. ''N''<sub>''S''</sub>(''G'') is defined as the set of all elements '''S''' ∈ ''S'' that map ''G'' onto itself by conjugation: |
− | ::: N<sub>S</sub>(G) := {'''S''' ∈S | '''S'''<sup> | + | ::: ''N''<sub>''S''</sub>(''G'') := {'''S''' ∈ ''S'' | '''S'''<sup>−1</sup>''GS'' = ''G''}. |
− | The normalizer N<sub>S</sub>(G) may coincide | + | The normalizer ''N''<sub>''S''</sub>(''G'') may coincide either 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 | + | == Euclidean vs affine normalizer == |
− | 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>E</sub>(G) := {'''S''' ∈ ''E'' | '''S'''<sup> | + | ::: ''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''. | ||
− | 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>A</sub>(G) := {'''S''' ∈ ''A'' | '''S'''<sup> | + | ::: ''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 | + | All symmetry operations of the Euclidean normalizer ''N''<sub>''E''</sub>(''G'') map the space group onto itself. The Euclidean normalizer 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''. |
== Euclidean normalizers of plane and space groups == | == Euclidean normalizers of plane and space groups == | ||
− | For all the plane / space groups except those corresponding to a [[pyroelectric point group]] the | + | For all the plane/space groups except those corresponding to a [[Pyroelectricity|pyroelectric point group]] the Euclidean normalizer is also a plane/space group. Instead, plane/space groups corresponding to a [[Pyroelectricity|pyroelectric point group]] have Euclidean normalizers that contain continuous translations in one, two or three independent lattice directions: these are not plane/space groups but supergroups of them. |
+ | == Euclidean normalizers of groups with specialized metric == | ||
+ | Plane/space groups where a specialized metric may induce a higher lattice symmetry have more than one type of Euclidean normalizer. This happens for 38 orthorhombic space groups (3 orthorhombic plane groups) as well as for the monoclinic and triclinic plane/space groups. | ||
+ | === Example === | ||
+ | A space group of the type ''Pmmm'' has three different Euclidean normalizers, all corresponding to basis vectors <math>\frac{1}{2}</math>'''a''',<math>\frac{1}{2}</math>'''b''',<math>\frac{1}{2}</math>'''c''': | ||
+ | * for the general case ''a'' ≠ ''b'' ≠ ''c'' ≠ ''a'', ''N''<sub>E</sub>(''Pmmm'') = ''Pmmm''; | ||
+ | * if ''a'' = ''b'' ≠ ''c'', N<sub>E</sub>(''Pmmm'') = ''P'' 4/''mmm''; | ||
+ | * if ''a'' = ''b'' = ''c'', N<sub>E</sub>(''Pmmm'') = <math>Pm(\bar 3)m</math>. | ||
+ | |||
+ | == Affine normalizers of plane and space groups == | ||
+ | The affine normalizer ''N''<sub>''A''</sub>(''G'') of a plane/space group ''G'' either is a true supergroup of the Euclidean normalizer of ''G'', ''N''<sub>''E''</sub>(''G''), or coincides with it: | ||
+ | |||
+ | ''N''<sub>''A''</sub>(''G'') ⊇ ''N''<sub>''E''</sub>(''G''). | ||
+ | |||
+ | Because any translation is an isometry, all translations belonging to ''N''<sub>''A''</sub>(''G'') also belong to ''N''<sub>''E''</sub>(''G''). Therefore, ''N''<sub>''A''</sub>(''G'') and ''N''<sub>''E''</sub>(''G'') necessarily have identical translation subgroups. | ||
+ | |||
+ | In contrast to the Euclidean normalizers, the affine normalizers of all plane/space groups are isomorphic groups: the type of the affine normalizer never depends on the metrical properties of the group ''G'', as is instead the case for the Euclidean normalizers. | ||
== See also == | == See also == | ||
− | Chapter | + | *[[Centralizer]] |
+ | *[[Stabilizer]] | ||
+ | *Chapter 3.5 of ''International Tables for Crystallography, Volume A'', 6th edition | ||
[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] |
Latest revision as of 13:12, 16 November 2017
Normaliseur (Fr). Normalisator (Ge). Normalizzatore (It). 正規化群 (Ja). Normalizador (Sp).
Contents
Definition
Given a group G and one of its supergroups S, they are uniquely related to a third, intermediated group N_{S}(G), called the normalizer of G with respect to S. N_{S}(G) is defined as the set of all elements S ∈ S that map G onto itself by conjugation:
- N_{S}(G) := {S ∈ S | S^{−1}GS = G}.
The normalizer N_{S}(G) may coincide either 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 E^{3} (or E^{2}) is called the Euclidean normalizer of G:
- N_{E}(G) := {S ∈ E | S^{−1}GS = 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 E^{3} (or E^{2}) is called the affine normalizer of G:
- N_{A}(G) := {S ∈ A | S^{−1}GS = G}.
'Symmetry of the symmetry pattern'
All symmetry operations of the Euclidean normalizer N_{E}(G) map the space group onto itself. The Euclidean normalizer 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.
Euclidean normalizers of plane and space groups
For all the plane/space groups except those corresponding to a pyroelectric point group the Euclidean normalizer is also a plane/space group. Instead, plane/space groups corresponding to a pyroelectric point group have Euclidean normalizers that contain continuous translations in one, two or three independent lattice directions: these are not plane/space groups but supergroups of them.
Euclidean normalizers of groups with specialized metric
Plane/space groups where a specialized metric may induce a higher lattice symmetry have more than one type of Euclidean normalizer. This happens for 38 orthorhombic space groups (3 orthorhombic plane groups) as well as for the monoclinic and triclinic plane/space groups.
Example
A space group of the type Pmmm has three different Euclidean normalizers, all corresponding to basis vectors [math]\frac{1}{2}[/math]a,[math]\frac{1}{2}[/math]b,[math]\frac{1}{2}[/math]c:
- for the general case a ≠ b ≠ c ≠ a, N_{E}(Pmmm) = Pmmm;
- if a = b ≠ c, N_{E}(Pmmm) = P 4/mmm;
- if a = b = c, N_{E}(Pmmm) = [math]Pm(\bar 3)m[/math].
Affine normalizers of plane and space groups
The affine normalizer N_{A}(G) of a plane/space group G either is a true supergroup of the Euclidean normalizer of G, N_{E}(G), or coincides with it:
N_{A}(G) ⊇ N_{E}(G).
Because any translation is an isometry, all translations belonging to N_{A}(G) also belong to N_{E}(G). Therefore, N_{A}(G) and N_{E}(G) necessarily have identical translation subgroups.
In contrast to the Euclidean normalizers, the affine normalizers of all plane/space groups are isomorphic groups: the type of the affine normalizer never depends on the metrical properties of the group G, as is instead the case for the Euclidean normalizers.
See also
- Centralizer
- Stabilizer
- Chapter 3.5 of International Tables for Crystallography, Volume A, 6th edition