Goursat's lemma

Groups

Goursat's lemma for groups can be stated as follows. : Let G, G' be groups, and let H be a subgroup of G\times G' such that the two projections p_1: H \to G and p_2: H \to G' are surjective (i.e., H is a subdirect product of G and G'). Let N be the kernel of p_2 and N' the kernel of p_1. One can identify N as a normal subgroup of G, and N' as a normal subgroup of G'. Then the image of H in G/N \times G'/N' is the graph of an isomorphism G/N \cong G'/N'. One then obtains a bijection between: :# Subgroups of G\times G' which project onto both factors, :# Triples (N, N', f) with N normal in G, N' normal in G' and f isomorphism of G/N onto G'/N'.

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 p_1: H \to G and p_2: H \to G' 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 H \leq p_1(H)\times p_2(H).

To motivate the proof, consider the slice S = {g} \times G' 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 (g,a), (g,b) \in S \cap H with a \in pN' \subset G' and b \in qN' \subset G', then H being a group, we get that (e, ab^{-1}) \in H, and hence, (e, ab^{-1}) \in N'. 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 G' \in G\times G' is exactly one particular coset of N' in G'. By an identical argument, the intersection of H with every "vertical" slice isomorphic to G \in G\times G' 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 G \times {e'} and {e} \times G', 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 h=(g,g') \in H, since p_1 is surjective. Therefore, p_1(N) is normal in G, viz: :gp_1(N) = p_1(h)p_1(N) = p_1(hN) = p_1(Nh) = p_1(N)g. It follows that N is normal in G \times {e'} since : (g,e')N = (g,e')(p_1(N) \times {e'}) = gp_1(N) \times {e'} = p_1(N)g \times {e'} = (p_1(N) \times {e'})(g,e') = N(g,e').

The proof that N' is normal in {e} \times G' proceeds in a similar manner.

Given the identification of G with G \times {e'}, we can write G/N and gN instead of (G \times {e'})/N 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 H \to G/N \times G'/N' defined by (g,g') \mapsto (gN, g'N'). The image of H under this map is {(gN,g'N') \mid (g,g') \in H }. Since H \to G/N is surjective, this relation is the graph of a well-defined function G/N \to G'/N' provided g_1N = g_2N \implies g_1'N' = g_2'N' for every (g_1,g_1'),(g_2,g_2') \in H, essentially an application of the vertical line test.

Since g_1N=g_2N (more properly, (g_1,e')N = (g_2,e')N), we have (g_2^{-1}g_1,e') \in N \subset H. Thus (e,g_2'^{-1}g_1') = (g_2,g_2')^{-1}(g_1,g_1')(g_2^{-1}g_1,e')^{-1} \in H, whence (e,g_2'^{-1}g_1') \in N', that is, g_1'N'=g_2'N'.

Furthermore, for every (g_1,g_1'),(g_2,g_2')\in H we have (g_1g_2,g_1'g_2')\in H. It follows that this function is a group homomorphism.

By symmetry, {(g'N',gN) \mid (g,g') \in H } is the graph of a well-defined homomorphism G'/N' \to G/N. 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.