Normalizer
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
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
Euclidean normalizers of groups with specialized metric
Plane / space groups where a specialized metric may induce a higher
Example
A space group of the type Pmmm has three different Euclidean normalizers, all corresponding to basis vectors a,b,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) = .
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 normalizer of all plane / space groups are isomorphic groups: the type of the affine normalizer never
See also
 Chapter 15 in the International Tables for Crystallography, Volume A