Goursat's lemma

Goursat's lemma, named after the French mathematician Édouard Goursat, is an algebraic theorem about subgroups of the direct product of two groups.

It can be stated more generally in a Goursat variety (and consequently it also holds in any Maltsev variety), from which one recovers a more general version of Zassenhaus' butterfly lemma. In this form, Goursat's theorem also implies the snake lemma.

Groups 
Goursat's lemma for groups 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 . One then obtains a bijection between :
 Subgroups of  which project onto both factors,
 Triples  with  normal in ,  normal in  and  isomorphism of  onto .

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

Notice that if  is any subgroup of  (the projections  and  need not be surjective), then the projections from  onto  and   are surjective. Then one can apply Goursat's lemma to .

To motivate the proof, consider the slice  in , for any arbitrary . By the surjectivity of the projection map to , this has a non trivial intersection with .  Then essentially, this intersection represents exactly one particular coset of .  Indeed, if we have elements  with  and , then  being a group, we get that , and hence, .  It follows that  and  lie in the same coset of .  Thus the intersection of  with every "horizontal" slice isomorphic to  is exactly one particular coset of  in .
By an identical argument, the intersection of  with every "vertical" slice isomorphic to  is exactly one particular coset of  in .

All the cosets of  are present in the group , and by the above argument, there is an exact 1:1 correspondence between them. The proof below further shows that the map is an isomorphism.

Proof 

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 . Since  is surjective, this relation is the graph of a well-defined function  provided  for every , essentially an application of the vertical line test.

Since  (more properly, ), we have . Thus , whence , that is, .

Furthermore, for every  we have . It follows that this function is a group homomorphism.

By symmetry,  is the graph of a well-defined homomorphism . These two homomorphisms are clearly inverse to each other and thus are indeed isomorphisms.

 Goursat varieties 

As a consequence of Goursat's theorem, one can derive a very general version on the Jordan–Hölder–Schreier theorem in Goursat varieties.

 References 
 Édouard Goursat, "Sur les substitutions orthogonales et les divisions régulières de l'espace", Annales Scientifiques de l'École Normale Supérieure (1889), Volume: 6, pages 9–102
 
 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.
 A. Carboni, G.M. Kelly and M.C. Pedicchio (1993), Some remarks on Mal'tsev and Goursat categories, Applied Categorical Structures, Vol. 4, 385-421.

Lemmas in group theory
Articles containing proofs