# Normal subgroup

### From Online Dictionary of Crystallography

Sous-groupe normal (*Fr*). Normalteiler (*Ge*). Sottogruppo normale (*It*). 正規部分群 (*Ja*). Subgrupo normal (*Sp*).

## Definition

A subgroup *H* of a group *G* is **normal** in *G* (*H* *G*) if *gH* = *Hg* for any *g* ∈ *G*. Equivalently, *H* ⊂ *G* is normal if and only if *gHg*^{−1} = *H* for any *g* ∈ *G*, *i.e.* if and only if each conjugacy class of *G* is either entirely inside *H* or entirely outside *H*. This is equivalent to saying that *H* is invariant under all inner automorphisms of *G*.

The property *gH* = *Hg* means that left and rights cosets of *H* in *G* coincide. From this one sees that the cosets
form a group with the operation *g*_{1}*H* * *g*_{2}*H* = *g*_{1}*g*_{2}*H* which is called
the factor group or **quotient group** of *G* by *H*, denoted by *G/H*.

In the special case that a subgroup *H* has only two cosets in *G* (namely *H* and *gH* for some *g* not contained in *H*), the subgroup *H* is always normal in *G*.

## Connection with homomorphisms

If *f* is a homomorphism from *G* to another group, then the kernel
of *f* is a normal subgroup of *G*. Conversely, every normal subgroup *H G* arises as the kernel of a homomorphism, namely of the projection homomorphism *G* → *G/H* defined by mapping *g* to its coset *gH*.

## Example

The group *T* containing all the translations of a space group *G* is a normal subgroup in *G* called the **translation subgroup** of *G*. The factor group *G/T* is isomorphic to the point group *P* of *G*.