Normal subgroup

In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element x in N and each g in G, the element g^-1xg is still in N. The statement N is a normal subgroup of G is written:

.

Another way to put this is saying that right and left cosets of N in G coincide:

N g = g g⁻¹ N g = g N    for all g in G.

A normal subgroup can also be defined by: A subgroup N of a group G is a normal subgroup if N is a union of conjugacy classes of G.

Normal subgroups are of relevance because if N is normal, then the factor group G/N may be formed. Normal subgroups of G are precisely the kernelss of group homomorphisms f : G → H.

{e} and G are always normal subgroups of G. If these are the only ones, then G is said to be simple.

All subgroups N of an abelian group G are normal, because g⁻¹Ng = g⁻¹gN = N.

See also:

• normalizer
• normal closure
• characteristic subgroup
• ideal (ring theory)