# Goursat's lemma

*Not to be confused with Goursat's integral lemma from Complex analysis*

**Goursat's lemma** is an algebraic theorem about subgroups of the direct product of two groups.

It can be stated as follows.

- Let , be groups, and let be a subgroup of such that the two projections and are surjective (i.e., is a subdirect product of and ). Let be the kernel of and the kernel of . One can identify as a normal subgroup of , and as a normal subgroup of . Then the image of in is the graph of an isomorphism .

An immediate consequence of this is that the subdirect product of two groups can be described as a fiber product and vice versa.

## Proof of Goursat's lemma

Before proceeding with the proof, and are shown to be normal in and , respectively. It is in this sense that and can be identified as normal in *G* and *G'*, respectively.

Since is a homomorphism, its kernel *N* is normal in *H*. Moreover, given , there exists , since is surjective. Therefore, is normal in *G*, viz:

It follows that is normal in since

The proof that is normal in proceeds in a similar manner.

Given the identification of with , we can write and instead of and , . Similarly, we can write and , .

On to the proof. Consider the map defined by . The image of under this map is . This relation is the graph of a well-defined function provided , essentially an application of the vertical line test.

Since (more properly, ), we have . Thus , whence , that is, . Note that by symmetry, it is immediately clear that , i.e., this function also passes the horizontal line test, and is therefore one-to-one. The fact that this function is a surjective group homomorphism follows directly.

## References

- Kenneth A. Ribet (Autumn 1976), "Galois Action on Division Points of Abelian Varieties with Real Multiplications",
*American Journal of Mathematics*, Vol. 98, No. 3, 751–804.