Normal subgroup
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 of which it is a part. In other words, a subgroup  of the group  is normal in  if and only if  for all  and  The usual notation for this relation is 

Normal subgroups are important because they (and only they) can be used to construct quotient groups of the given group.  Furthermore, the normal subgroups of  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  of a group  is called a normal subgroup of  if it is invariant under conjugation; that is, the conjugation of an element of  by an element of  is always in   The usual notation for this relation is

Equivalent conditions
For any subgroup  of  the following conditions are equivalent to  being a normal subgroup of  Therefore, any one of them may be taken as the definition:

 The image of conjugation of  by any element of  is a subset of 
 The image of conjugation of  by any element of  is equal to 
 For all  the left and right cosets  and  are equal.
 The sets of left and right cosets of  in  coincide.
 The product of an element of the left coset of  with respect to  and an element of the left coset of  with respect to  is an element of the left coset of  with respect to : for all  if and  then 
  is a union of conjugacy classes of 
  is preserved by the inner automorphisms of 
 There is some group homomorphism  whose kernel is 
 For all  and  the commutator  is in 
 Any two elements commute regarding the normal subgroup membership relation: for all   if and only if

Examples 
For any group  the trivial subgroup  consisting of just the identity element of  is always a normal subgroup of   Likewise,  itself is always a normal subgroup of  (If these are the only normal subgroups, then  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.

If  is an abelian group then every subgroup  of  is normal, because  A group that is not abelian but for which every subgroup is normal is called a Hamiltonian group.

A concrete example of a normal subgroup is the subgroup  of the symmetric group  consisting of the identity and both three-cycles.  In particular, one can check that every coset of  is either equal to  itself or is equal to   On the other hand, the subgroup  is not normal in  since  This illustrates the general fact that any subgroup  of index two is normal.

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 

 If  is a normal subgroup of  and  is a subgroup of  containing  then  is a normal subgroup of 
 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.
 The two groups  and  are normal subgroups of their direct product 
 If the group  is a semidirect product  then  is normal in  though  need not be normal in 
 Normality is preserved under surjective homomorphisms; that is, if  is a surjective group homomorphism and  is normal in  then the image  is normal in 
 Normality is preserved by taking inverse images; that is, if  is a group homomorphism and  is normal in  then the inverse image  is normal in 
 Normality is preserved on taking direct products; that is, if  and  then 
 Every subgroup of index 2 is normal. More generally, a subgroup,  of finite index,  in  contains a subgroup,  normal in  and of index dividing  called the normal core. In particular, if  is the smallest prime dividing the order of  then every subgroup of index  is normal.
 The fact that normal subgroups of  are precisely the kernels of group homomorphisms defined on  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 
Given two normal subgroups,  and  of  their intersection and their product  are also normal subgroups of  

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

The lattice is complete and modular.

Normal subgroups, quotient groups and homomorphisms 

If  is a normal subgroup, we can define a multiplication on cosets as follows: 

This relation defines a mapping  To show that this mapping is well-defined, one needs to prove that the choice of representative elements  does not affect the result. To this end, consider some other representative elements  Then there are  such that  It follows that where we also used the fact that  is a  subgroup, and therefore there is  such that  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 and denoted with  There is a natural homomorphism,  given by  This homomorphism maps  into the identity element of  which is the coset  that is, 

In general, a group homomorphism,  sends subgroups of  to subgroups of  Also, the preimage of any subgroup of  is a subgroup of  We call the preimage of the trivial group  in  the kernel of the homomorphism and denote it by  As it turns out, the kernel is always normal and the image of  is always isomorphic to  (the first isomorphism theorem). In fact, this correspondence is a bijection between the set of all quotient groups of  and the set of all homomorphic images of  (up to isomorphism). It is also easy to see that the kernel of the quotient map,  is  itself, so the normal subgroups are precisely the kernels of homomorphisms with domain

See also

Operations taking subgroups to subgroups
Normalizer
Conjugate 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)

Notes

References

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 Abstract Algebra. The Basic Graduate Year
 Timothy Gowers, Normal subgroups and quotient groups
 John Baez, What's a Normal Subgroup?

Subgroup properties