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
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 ringss.