Normal subgroup

A subgroup H of a group G is normal in G (H $\triangleleft$ 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 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 Hdenoted 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 $\triangleleft$ 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.