Encyclopedia > Fundamental theorem on homomorphisms

  Article Content

Fundamental theorem on homomorphisms

For some 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:

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

All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article
BBC News 24

... with so few viewers. Their response was to promote the channel through their ordinary channels BBC1 and BBC2, using terrestrial signals, and this is seen by some as ...

This page was created in 34.4 ms