Group homomorphism

In group theory one considers special maps between groups, called group homomorphisms. A group homomorphism mapping between two groups which is compatible therewith, and therefore a special homomorphism.

Definition

Given two groups A function is called a homomorphism if and only if for all elements:

The equation says that the homomorphism is structure preserving: It does not matter if you only connects two elements and maps the result or whether it reflects only the two elements and then combines the images.

From this definition it follows that a group homomorphism is the neutral element of the neutral element of maps:

As is true for all

Thus, the neutral element is.

Furthermore, it follows that he reflects on Inverse Inverse:

Because because

Is the inverse of

Image and core

As an image ( engl. image ) of the group homomorphism refers to the amount of image below:

The core ( engl. kernel ) is the archetype of the neutral element:

Just when true (the core of so only the neutral element of contains, which is always in the core ) is injective. An injective homomorphism is also called group monomorphism.

The core of is always a normal subgroup of and the image of is a subgroup of. After the homomorphism the factor group is isomorphic to.

Examples

  • Consider the additive group of integers and the factor group. The figure (see congruence and residue class ring) is a group homomorphism. It is injective, and its core consists of the set of all integers is divisible by 3. This homomorphism is called the canonical projection.
  • The exponential function is a group homomorphism between the additive group of real numbers and the multiplicative group of real numbers not equal to 0, because. This map is injective, and its image is the set of positive real numbers.
  • The complex exponential function is a group homomorphism between the complex numbers with addition and different from 0 complex numbers with multiplication. This homomorphism is surjective and its kernel is, as can be seen from the Euler's identity, for example.
  • Are and any groups, then the map that maps each element to the identity element of a group homomorphism. Its core is all about.
  • For each group is the identity map, a bijective homomorphism.

Concatenation of group homomorphisms

Are and two group homomorphisms, then their composition is also a group homomorphism.

The class of all groups with group homomorphisms forms a category.

A homomorphism is called

  • Monomorphism if it is injective.
  • Epimorphism if it is surjective.
  • Isomorphism if it is bijective.

Is a group isomorphism, then its inverse function is a group isomorphism, the groups and are then called isomorphic to each other: They differ only in the name of their elements and vote for almost all purposes the same.

Is a group homomorphism of a group in themselves, then it is called Gruppenendomorphismus. Is he beyond bijective, then it is called Gruppenautomorphismus. The set of all Gruppenendomorphismen of forms composing a monoid. The set of all Gruppenautomorphismen a group formed with the composition of a group, the automorphism group of.

The automorphism of only includes two elements: the identity ( 1 ), and the multiplication by -1; it is isomorphic to the cyclic group.

In the group of each linear map with an automorphism.

Homomorphisms between abelian groups

If G and H abelian (ie commutative ) groups, then, the set Hom (G, H ) of all group homomorphisms from G to H itself a (again abelian ) group, namely with the " pointwise addition":

The commutativity of H one needs in order h k is a group homomorphism again.

The set of endomorphisms of an abelian group G forms, with the addition of a group that is called End ( G).

The addition of homomorphisms is in the following sense compatible with the composition: Are f in Hom (K, G), h, k in Hom (G, H ), g in Hom (H, L ), then

This shows that the Endomorphismengruppe End ( G) of an abelian group even forms a ring, the endomorphism ring of G.

For example, the endomorphism ring of the Klein four-group is isomorphic to the ring of 2 × 2 matrices over the residue field.

Algebraic structure Fundamental theorem on homomorphisms Quotient group Modular arithmetic Matrix ring
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