Normal subgroup

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

gH = Hg means that left and rights cosets of H in G coincide. As a consequence, every subgroup with only one other coset is normal.

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.