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 lemma also implies the snake lemma.

Groups

Goursat's lemma for groups can be stated as follows. 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 H is any subgroup of G\times G' (the projections and need not be surjective), then the projections from H onto p_1(H) and p_2(H) are surjective. Then one can apply Goursat's lemma to. To motivate the proof, consider the slice in G \times G', for any arbitrary g \in G. By the surjectivity of the projection map to G, this has a non trivial intersection with H. Then essentially, this intersection represents exactly one particular coset of N'. Indeed, if we have elements with and, then H being a group, we get that , and hence,. It follows that (g,a) and (g,b) lie in the same coset of N'. Thus the intersection of H with every "horizontal" slice isomorphic to is exactly one particular coset of N' in G'. By an identical argument, the intersection of H with every "vertical" slice isomorphic to is exactly one particular coset of N in G. All the cosets of N,N' are present in the group H, 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, N and N' are shown to be normal in and, respectively. It is in this sense that N and N' can be identified as normal in G and G', respectively. Since p_2 is a homomorphism, its kernel N is normal in H. Moreover, given g \in G, there exists, since p_1 is surjective. Therefore, p_1(N) is normal in G, viz: It follows that N is normal in since The proof that N' is normal in proceeds in a similar manner. Given the identification of G with, we can write G/N and gN instead of and (g,e')N, g \in G. Similarly, we can write G'/N' and g'N', g' \in G'. On to the proof. Consider the map defined by. The image of H under this map is. Since H \to G/N is surjective, this relation is the graph of a well-defined function provided for every, essentially an application of the vertical line test. Since g_1N=g_2N (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.