Normal subgroup

In mathematics, more specifically in abstract algebra, a normal subgroup is a special kind of subgroup. Normal subgroups are important because they can be used to construct quotient groups from a given group. The analogous concept in ring theory is called ideal.

Évariste Galois was the first to realize the importance of the existence of normal subgroups.

Definitions

A subgroup N of a group G is called normal subgroup if it is invariant under conjugation; that is, for each element n in N and each g in G, the element gng⁻¹ is still in N. We write

N\triangleleft G.

The following conditions are equivalent to requiring that a subgroup N be normal in G. Any one of them may be taken as the definition:

For all g in G, gNg⁻¹ ⊆ N.
For all g in G, gNg⁻¹ = N.
The sets of left and right cosets of N in G coincide.
For each g in G, gN = Ng.
N is a union of conjugacy classes of G.
There is some homomorphism on G for which N is the kernel.

Note that condition (1) is logically weaker than condition (2), and condition (3) is logically weaker than condition (4). For this reason, conditions (1) and (3) are often used to prove that N is normal in G, while conditions (2) and (4) are used to prove consequences of the normality of N in G.

Examples

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

The center of a group is a normal subgroup.

The commutator subgroup is a normal subgroup.

All subgroups N of an abelian group G are normal, because gN = Ng. A group that is not abelian but for which every subgroup is normal is called a Hamiltonian group.

The translation group in any dimension is a normal subgroup of the Euclidean group; for example in 3D rotating, translating, and rotating back results in only translation; also reflecting, translating, and reflecting again results in only translation (a translation seen in a mirror looks like a translation, with a reflected translation vector). The translations by a given distance in any direction form a conjugacy class; the translation group is the union of those for all distances.

Properties

Normality is preserved upon surjective homomorphisms, and is also preserved upon taking inverse images.
Normality is preserved on taking direct products
A normal subgroup of a normal subgroup need not be normal. That is, normality is not a transitive relation. However, a characteristic subgroup of a normal subgroup is normal. Also, a normal subgroup of a central factor is normal. In particular, a normal subgroup of a direct factor is normal.
Every subgroup of index 2 is normal. More generally, a subgroup H of finite index n in G contains a subgroup K normal in G and of index dividing n!.
Even more generally, if p is the smallest prime dividing the order of G, then every subgroup of index p is normal.

Lattice of normal subgroups

The normal subgroups of a group G form a lattice under subset inclusion with least element {e} and greatest element G. Given two normal subgroups N and M in G, meet is defined as

N \wedge M := N \cap M

and join is defined as

N \vee M := N M = \{nm \,|\, n \in N \,, m \in M\}

Normal subgroups and homomorphisms

Normal subgroups are of relevance because if N is normal, then the quotient group G/N may be formed: if N is normal, we can define a multiplication on cosets by

(a₁N)(a₂N) := (a₁a₂)N

This turns the set of cosets into a group called the quotient group G/N. There is a natural homomorphism f : G → G/N given by f(a) = aN. The image f(N) consists only of the identity element of G/N, the coset eN = N.

In general, a group homomorphism f: G → H sends subgroups of G to subgroups of H. Also, the preimage of any subgroup of H is a subgroup of G. We call the preimage of the trivial group {e} in H the kernel of the homomorphism and denote it by ker(f). As it turns out, the kernel is always normal and the image f(G) of G is always isomorphic to G/ker(f). In fact, this correspondence is a bijection between the set of all quotient groups G/N of G and the set of all homomorphic images of G (up to isomorphism). It is also easy to see that the kernel of the quotient map, f: G → G/N, is N itself, so we have shown that the normal subgroups are precisely the kernels of homomorphisms with domain G.

References

I. N. Herstein, Topics in algebra. Second edition. Xerox College Publishing, Lexington, Mass.-Toronto, Ont., 1975. xi+388 pp.
David S. Dummit; Richard M. Foote, Abstract algebra. Prentice Hall, Inc., Englewood Cliffs, NJ, 1991. xiv+658 pp. ISBN 0-13-004771-6