Isomorphism Theorem
The isomorphism theorems are two mathematical theorems that make statements about groups . They can also be transferred to more complex algebraic structures and are therefore an important result of universal algebra . The isomorphism theorems are a direct consequence of the homomorphism theorem of the corresponding algebraic structure .
Sometimes the homomorphism set is called the first isomorphism set. The sentences given below are then called the second or third isomorphism sentence .
Group theory
First isomorphism theorem
Let there be a group, a normal subgroup in and a subgroup of . Then the complex product is also a subgroup of , is a normal part of and the group is normal part of in . The following applies:
The isomorphism of groups denotes .
The isomorphism that is usually meant is called canonical isomorphism. According to the homomorphism theorem, it is derived from the surjective mapping
induced, because it obviously holds
- .
From the first isomorphism theorem, as a special case, one receives the clear statement that one can "expand" with exactly when .
Second isomorphism theorem
Let there be a group, a normal subgroup in and a subgroup of which is normal subgroup in . Then:
In this case canonical isomorphisms can be given in both directions, induced by on the one hand
on the other hand through
To put it clearly, the second isomorphism statement says that one can "shorten".
Rings
The isomorphism theorems also apply to rings in an adapted form:
First isomorphism theorem
Let it be a ring, an ideal of and a sub-ring of . Then the sum is a subring of and the cut is an ideal of . The following applies:
The isomorphism of rings denotes .
Second isomorphism theorem
Let it be a ring, two ideals of . Then is an ideal of . The following applies:
Vector spaces, Abelian groups, or objects of any Abelian category
Be there
- Vector spaces over a body
- or Abelian groups
- or more generally modules over a ring
- or objects of an Abelian category in general .
Then:
Here, too, the symbol stands for the isomorphism of the corresponding algebraic structures or objects in the respective category.
The canonical isomorphisms are clearly determined by the fact that they are compatible with the two canonical arrows of or .
The serpent lemma provides a far-reaching generalization of the isomorphism theorems .
literature
- Siegfried Bosch : Algebra. 8th edition. Springer, Berlin / Heidelberg 2013, ISBN 978-3-642-39566-6 , chapter 1.2.
- Christian Karpfinger, Kurt Meyberg: Algebra. 3. Edition. Springer, Berlin / Heidelberg 2013, ISBN 9783827430113 , chapter 4.6.
Web links
matheplanet.com: Peer pressure IV - Detailed explanations and proofs of the isomorphism theorems