Isomorphism theorem

The isomorphism theorems are two mathematical sentences that make statements about groups. They can be transferred to more complex algebraic structures and are thus an important result of universal algebra. The isomorphism theorems are a direct consequence of the homomorphism of the corresponding algebraic structure.

Sometimes the homomorphism is called the first isomorphism theorem. The rates shown below are then called accordingly second and third isomorphism theorem.

Group Theory

First isomorphism Theorem

There were a group is a normal subgroup and in a subgroup of. Then the product is a subgroup of a normal subgroup is in and the group is a normal subgroup in. The following applies:

It denotes the isomorphism of groups.

The isomorphism, which is usually meant is called the canonical isomorphism. He is under the homomorphism of the surjective map

Induced, because it seems to be true

To put it clearly implies the first isomorphism theorem that one can " expand " with N.

Second isomorphism Theorem

There were a group is a normal subgroup and in a subgroup of the normal subgroup in. Then:

In this case you can specify canonical isomorphisms in both directions, on the one hand induced by

On the other hand by

To put it clearly implies the second isomorphism theorem that one can " cut " N.

Vector spaces, abelian groups, or objects of any abelian category

There are

• Vector spaces over a field
• Or abelian groups
• Or more generally modules over a ring
• Or, more generally, objects of an abelian category.

Then:

Here is the symbol for the isomorphism of the corresponding algebraic structures or objects in each category.

The canonical isomorphisms are uniquely determined by the fact that they are compatible with the two canonical arrows of or.

A far-reaching generalization of the isomorphism theorems provides the Schlangenlemma.