Normalizer

Normaliseur (Fr). Normalisator (Ge). Normalizzatore (It). 正規化群 (Ja). Normalizador (Sp).

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⁻¹GS = 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 E³ (or E²) is called the Euclidean normalizer of G:

N_E(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:

N_A(G) := {S ∈ A | S⁻¹GS = 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.