# Normal subgroup

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 say that H is invariant under all inner automorphisms of G.

The property gH = Hg means that left and rights cosets of H in G *g _{1}H * g_{2}H = g_{1}g_{2}H* which is called the factor group or

**of**quotient group

*G*by

*H*,

*G/H*.

In the *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 *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.