# Normalizer

### From Online Dictionary of Crystallography

##### Revision as of 18:59, 24 February 2007 by MassimoNespolo (talk | contribs)

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**^{-1}GS = G}

- N

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 E^{3} (or E^{2}) is called the *Euclidean normalizer of G*:

- N
_{S}(G) := {**S**∈*E*|**S**^{-1}G**S**= G}

- N

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
_{S}(G) := {**S**∈*A*|**S**^{-1}GS = G}

- N

## See also

Chapter 15 in the *International Tables for Crystallography, Volume A*