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]]