Normalizer

Normaliseur (Fr); Normalizzatore (It).

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 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:

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 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.

Euclidean normalizers of plane and space groups

For all the plane / space groups except those corresponding to a pyroelectric point group the Euclidan normalizer is also a plane / space group. Instead, plane / space groups corresponding to a pyroelectric point group have Euclidean normalizers that contain contrinuous translations in one, two or three indepenedent lattice directions: these are not plane / space groups but supergroupes of them.

See also

Chapter 15 in the International Tables for Crystallography, Volume A