Normal subgroup

Sous-groupe normal (Fr). Sottogruppo normale (It). 正規部分群 (Ja).

Definition

A subgroup H of a group G is normal in G (H [math]\triangleleft[/math] 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 g₁H * g₂H = g₁g₂H 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 [math]\triangleleft[/math] 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.