In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group

Normal subgroups are important because they (and only they) can be used to construct quotient groups are precisely the kernels of group homomorphisms with domain which means that they can be used to internally classify those homomorphisms.

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

Definitions

A subgroup is called a normal subgroup if it is invariant under conjugation

Equivalent conditions

the following conditions are equivalent Therefore, any one of them may be taken as the definition.

• .

• .

• the left and right cosets are equal.

• The sets of left and right cosets coincide.

• . (This group is called the quotient group modulo.)

• is a union of conjugacy classes

• is preserved by the inner automorphisms

• There is some group homomorphism whose kernel

• whose fibers mentioned above.)

• There is some congruence relation for which the equivalence class of the identity element.

• the commutator

Examples

is said to be simple.) Other named normal subgroups of an arbitrary group include the center of the group (the set of elements that commute with all other elements) and the commutator subgroup More generally, since conjugation is an isomorphism, any characteristic subgroup is a normal subgroup.

is an abelian group, every subgroup of the center, hence the fact that all subgroups of an abelian group are normal.) A group that is not abelian but for which every subgroup is normal is called a Hamiltonian group.

of the symmetric group of index two is normal.

As an example of a normal subgroup within a matrix group, consider the general linear group matrices of determinant 1 (the special linear group, so it is a normal subgroup.

In the Rubik's Cube group, the subgroups consisting of operations which only affect the orientations of either the corner pieces or the edge pieces are normal.

The translation group is a normal subgroup of the Euclidean group in any dimension. This means: applying a rigid transformation, followed by a translation and then the inverse rigid transformation, has the same effect as a single translation. By contrast, the subgroup of all rotations about the origin is not a normal subgroup of the Euclidean group, as long as the dimension is at least 2: first translating, then rotating about the origin, and then translating back will typically not fix the origin and will therefore not have the same effect as a single rotation about the origin.

Properties

• A normal subgroup of a normal subgroup of a group need not be normal in the group. That is, normality is not a transitive relation. The smallest group exhibiting this phenomenon is the dihedral group of order 8. However, a characteristic subgroup of a normal subgroup is normal. A group in which normality is transitive is called a T-group.

• are normal subgroups of their direct product

• is a semidirect product

• Normality is preserved under surjective homomorphisms;

• Normality is preserved by taking inverse images;

• Normality is preserved on taking direct products;

• Every subgroup of index called the normal core is normal.

• accounts for some of the importance of normal subgroups; they are a way to internally classify all homomorphisms defined on a group. For example, a non-identity finite group is simple if and only if it is isomorphic to all of its non-identity homomorphic images, a finite group is perfect if and only if it has no normal subgroups of prime index, and a group is imperfect if and only if the derived subgroup is not supplemented by any proper normal subgroup.

Lattice of normal subgroups

and their product

form a lattice under subset inclusion with least element and greatest element The meet in this lattice is their intersection and the join is their product.

The lattice is complete and modular.

Normal subgroups, quotient groups and homomorphisms

is a This proves that this product is a well-defined mapping between cosets.

With this operation, the set of cosets is itself a group, called the quotient group There is a natural homomorphism

the kernel is always isomorphic (the first isomorphism theorem). (up to isomorphism). itself, so the normal subgroups are precisely the kernels of homomorphisms with domain

See also

Operations taking subgroups to subgroups

• Normalizer

• Normal closure

• Normal core

Subgroup properties complementary (or opposite) to normality

• Malnormal subgroup

• Contranormal subgroup

• Abnormal subgroup

• Self-normalizing subgroup

Subgroup properties stronger than normality

• Characteristic subgroup

• Fully characteristic subgroup

Subgroup properties weaker than normality

• Subnormal subgroup

• Ascendant subgroup

• Descendant subgroup

• Quasinormal subgroup

• Seminormal subgroup

• Conjugate permutable subgroup

• Modular subgroup

• Pronormal subgroup

• Paranormal subgroup

• Polynormal subgroup

• C-normal subgroup

Related notions in algebra

• Ideal (ring theory)

• Semigroup ideal

Notes

References

Bibliography

Further reading

• I. N. Herstein, Topics in algebra. Second edition. Xerox College Publishing, Lexington, Mass.-Toronto, Ont., 1975. xi+388 pp.

External links

• Normal subgroup in Springer's Encyclopedia of Mathematics

• Robert Ash: Group Fundamentals in

• Timothy Gowers, Normal subgroups and quotient groups

• John Baez, What's a Normal Subgroup?