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 π be the natural surjective homomorphism G->G/K. Then there is an injective homomorphism h:G/K->K such that f = h π. Moreover, G/K is isomorphic to Im f (the image of f).
The situation is described by the (ugly, ASCII) diagram:
f
G --> H
| 7
pi | / h
V /
G/K