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
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
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.