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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 kernels 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)In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. For instance, in rings one studies prime ideals instead of prime numbers, one defines coprime ideals as a generalizati