5. The correspondence theorem

One version for groups: If $\varphi: G \rightarrow H$ is a surjective homomorphism, then there is a one-to-one correspondence between the normal subgroups of $H$ and the normal subgroups of $G$ that contain ker$\; \varphi$. In fact, the subgroup of $G$ corresponding to a normal subgroup $K$ of $H$ is simply $\varphi ^ {-1}(K)$.

The generalization to algebras is this:

5.1 Theorem (correspondence theorem). If $\varphi: {\cal A} \rightarrow {\cal B}$ is a surjective homomorphism, then there is a one-to-one correspondence between congruence relations on ${\cal B}$ and the congruence relations on ${\cal A}$ that contain ker$\; \varphi$.

5.2 Note. Using the first isomorphism theorem, equivalent versions can be given for the natural maps $G\rightarrow G/N$ (where $N \, \triangleleft \; G$) or ${\cal A} \rightarrow {\cal A}/\theta$ (where $\theta \in$   Con$({\cal A})$).

5.3 Note. For groups, one can also say that there is a one-to-one correspondence between all subgroups of $H$, normal or not, and those subgroups of $G$ that contain ker$\; \varphi$. For algebras in general, this becomes a statement about subalgebras rather than about congruence relations.