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*.

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