Normal subgroup

In group theory, a normal subgroup or a normal subgroup is a special subset of a group. With the help of factor groups of the group can be formed.

This allows the structure analysis of groups are reduced to less complex groups. Example, you can describe all possible homomorphisms from a group essentially when the normal subgroup of the group are known.

The French mathematician Galois Évariste recognized in the nineteenth century as first the importance of the concept of " normal subgroup " for the study of non-commutative groups.

• 4.1 Complementary normal subgroup and internal direct product
• 4.2 Internal semidirect product

Definition

Stands for the sub-group to be viewed by and for an arbitrary element of. The left coset of according to the element of is a subset of :; accordingly represents a right coset.

The subgroup is called a normal subgroup of, if one of the following conditions is met, which are pairwise equivalent:

It is also said that the subgroup is normal in; the terms normal subgroup and normal subgroup are identical. The notation means " is a normal subgroup of ."

Comments

Each group has himself and consisting only of the identity element one sub-group as a normal subgroup. Groups, which themselves have no normal subgroups other than the one- group and simply hot. The classification problem for finite groups can be attributed mainly to the classification of finite simple groups, which is in the twentieth century completely successful.

Every subgroup of an abelian group is a normal subgroup of the group, and many statements about normal subgroup is trivial for commutative groups. The center and the commutator subgroup of a group are always normal subgroups.

The normal subgroup relation is not transitive, that is, and does not follow in general.

A sub-group if and only normal subgroups in if you normalizer is all about. A subgroup is always normal subgroup in its normalizer.

All the characteristic subgroups of a group are normal subgroups of the group, because the conjugation of group elements is an automorphism. The converse is not true in general.

A subgroup of index 2 is always a normal subgroup. Is a subset of the index and is the smallest prime number that divides the order of, as a normal divider.

Normal subgroups, group homomorphisms and factor group

Every normal subgroup of a group is the core of a group homomorphism from the group, that is, the image below is the neutral element. As a homomorphism, the canonical surjective projection

Are elected by the factor group, which assigns to each group element has its coset.

Factor group

The cosets of normal subgroups form by a group factor group of the complex product.

Specifically, this means: The factor group consists of the cosets. The product is defined by the complex product. This raises the question as to whether this product is well-defined. To check this, choose two representatives and the same coset. It is then so

And analogously

If the complex product is well-defined, so it must be independent of the choice of representatives to be, so it must be true:

If you add the above product, it can be seen that the above equality holds only if N is a normal subgroup of the group. In summary, therefore: If and only if N is a normal subgroup of the group arises as a complex product of any two cosets again a coset of N, and it is. Just then, can this complex product that is always determined by multiplication of two representatives of the cosets and subsequent coset formation.

Homomorphism and isomorphism theorem first

The core of any group homomorphism is always a normal subgroup of the group abgebildeteten. This is a consequence of the consideration of the above section, but can also be proved directly. to this end let

Then for all and

So and so is a normal subgroup in by Definition 2

Each group homomorphism induces an isomorphism

Of the factor group on the image. This statement is often called a homomorphism ( for groups). Is the homomorphism is surjective, then the factor group after the core is isomorphic to the image.

Are and subgroups of the group and is even a normal subgroup of, then applies

This statement is often referred to as the first isomorphism theorem ( for groups).

Normal subgroup and subgroup association

The normal subgroups of a group form a system of sets, which is even a cover system. This containment system is a complete lattice, the normal subgroup Association. Here this means:

In addition, the normal subgroup lattice is a modular sublattice of the subgroup Association.

In the following two sections must be distinguished in the considered product groups between outer and inner products, although the two versions always have the same algebraic structure ( isomorphic ). The formal differences between inner and outer products are discussed in the article on the semi- direct product and in this article at the end of the section on the semi- direct product.

Complementary normal subgroup and internal direct product

In general there is no complementary properties in the normal subgroup Association. Has a normal subgroup, however, a complementary object, that is, valid for the normal subgroup and then the group can be represented as (internal ) direct product of normal subgroups: that is, each group member has a unique representation as a product of elements and. Conversely, any factor of a ( external ) direct product ( isomorphic to a ) normal subgroup of the product group and the product of the remaining factors is isomorphic to a complementary normal subgroup.

A generalization of this statement: For two normal subgroups that have a trivial intersection, that is, the following applies:

• Its elements commute with each other, without, of course, one of the two normal subgroups should be commutative:

• Your supremum of the Association of normal subgroups is consistent with their complex product, which in turn is isomorphic to its ( external ) direct product:

Both statements are true in general for subgroups which are not normal subgroups not to. For example, overlap in the free group on two elements, the two infinite cyclic subgroups and in the one group. The group ( external direct product ) is, however, no subgroup of isomorphic. The complex product is not a subset of, since, for example, but.

Interior semidirect product

Is just a normal subgroup and not necessary normal subgroup of the group and cut the two in one group, so true, then:

• The complex product is a (not necessarily normal ) subgroup of.
• Each element is represented as a product of elements and unique.
• Of course, the normal subgroup of always is normal in. The subgroup is normal if and only in when the elements of and commute with each other (see above).

In the described situation ( ) is called the complex product as (internal ) semidirect product of the subgroups and. The outer semidirect product consists, as stated in that article, from the Cartesian product of two groups (here and ) together with a homomorphism of the group of automorphisms of. The outer semi- direct product is then often written as. From the technical details of interest in our context only that the calculation rule ( relation):

Is introduced to the Cartesian product. The notation here means the automorphism is applied to, it applies here as in the following always. This calculation rule makes it possible to bring all products ( by pushing through the elements of to the right) to the standard form. In our case, an inner product corresponds to the calculation rule

That is operated on by conjugation, is defined by this conjugation automorphism of the normal subgroup. For the purposes of these considerations, the complex product ( here an inner semidirect product ) is isomorphic to the outer semi- direct product.

Every direct product is also a special semidirect, as here described is exactly then the (internal ) direct product of and, if one of the following equivalent conditions is true:

• (also is a normal subgroup of the product),
• ( Elements of the two factor groups can be swapped in products among themselves without the value of the product changes )
• (Conjugation with elements of leaves pointwise fixed).