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_S(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_S(G) := {S ∈ A | S^-1GS = G}

See also

Chapter 15 in the International Tables for Crystallography, Volume A