meanings of Fundamental theorem on homomorphisms definition of Fundamental theorem on homomorphisms books about Fundamental theorem on homomorphisms references on Fundamental theorem on homomorphisms articles about Fundamental theorem on homomorphisms dreams about Fundamental theorem on homomorphisms
 Fundamental theorem on homomorphisms - Definition 

In abstract algebra, for a number of algebraic structures, the fundamental theorem on homomorphisms relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism.

For groups, the theorem states:

Let G and H be groups; let f : G->H be a group homomorphism; let K be the kernel of f; let φ be the natural surjective homomorphism G->G/K. Then there exists a unique homomorphism h:G/K->H such that f = h φ. Moreover, h is injective and provides an isomorphism between G/K and the image of f.

The situation is described by the following commutative diagram:

image:FundHomDiag.png

Similar theorems are valid for monoids, vector spaces, modules, and rings.

de:Homomorphiesatz es:Teorema fundamental sobre homomorfismos

Copyright 2008 WordIQ.com - Privacy Policy  ::  Terms of Use  :: Contact Us  :: About Us
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Fundamental theorem on homomorphisms".