Group isomorphism

A group isomorphism is a mathematical object from the algebra, which is especially seen in group theory. Similarly to other definitions of isomorphisms of group isomorphism is defined as a bijective homomorphism. A group isomorphism which maps a group to itself, is a Gruppenautomorphismus.

Applications can be found for example in the Gruppenisomorphismen Isomorphiesätzen.

Definition

Let and be two groups. A group homomorphism group isomorphism is called if an inverse mapping has, that is, if it with a group homomorphism and there. Equivalent to this is the requirement that the group homomorphism is bijective.

Makes the group isomorphism from to from are known as domain and image set equal, it is called the group isomorphism also Gruppenautomorphimus.

Properties

• As a group isomorphism is injective, its core consists only of the identity element:

• His image is the whole group, that is:

• For each group isomorphism exists a uniquely determined inverse function.

Isomorphism of groups

Groups, between which there exists such a group isomorphism is called isomorphic to each other: they differ only in the name of their elements and vote for almost all purposes the same.

It is easy to show that the isomorphism of groups is an equivalence relation.

Examples

• For each group G is the identity map, a Gruppenautomorphismus.
• The exponential function is a group isomorphism.
• The conjugation describes a Gruppenautomorphismus.