Goursat's lemma

Goursat's lemma, named after the France 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.

Let [math]\displaystyle{ G }[/math], [math]\displaystyle{ G' }[/math] be groups, and let [math]\displaystyle{ H }[/math] be a subgroup of [math]\displaystyle{ G\times G' }[/math] such that the two projections [math]\displaystyle{ p_1: H \to G }[/math] and [math]\displaystyle{ p_2: H \to G' }[/math] are surjective (i.e., [math]\displaystyle{ H }[/math] is a subdirect product of [math]\displaystyle{ G }[/math] and [math]\displaystyle{ G' }[/math]). Let [math]\displaystyle{ N }[/math] be the kernel of [math]\displaystyle{ p_2 }[/math] and [math]\displaystyle{ N' }[/math] the kernel of [math]\displaystyle{ p_1 }[/math]. One can identify [math]\displaystyle{ N }[/math] as a normal subgroup of [math]\displaystyle{ G }[/math], and [math]\displaystyle{ N' }[/math] as a normal subgroup of [math]\displaystyle{ G' }[/math]. Then the image of [math]\displaystyle{ H }[/math] in [math]\displaystyle{ G/N \times G'/N' }[/math] is the graph of an isomorphism [math]\displaystyle{ G/N \cong G'/N' }[/math]. One then obtains a bijection between:

1. Subgroups of [math]\displaystyle{ G\times G' }[/math] which project onto both factors,
2. Triples [math]\displaystyle{ (N, N', f) }[/math] with [math]\displaystyle{ N }[/math] normal in [math]\displaystyle{ G }[/math], [math]\displaystyle{ N' }[/math] normal in [math]\displaystyle{ G' }[/math] and [math]\displaystyle{ f }[/math] isomorphism of [math]\displaystyle{ G/N }[/math] onto [math]\displaystyle{ G'/N' }[/math].

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 [math]\displaystyle{ H }[/math] is any subgroup of [math]\displaystyle{ G\times G' }[/math] (the projections [math]\displaystyle{ p_1: H \to G }[/math] and [math]\displaystyle{ p_2: H \to G' }[/math] need not be surjective), then the projections from [math]\displaystyle{ H }[/math] onto [math]\displaystyle{ p_1(H) }[/math] and [math]\displaystyle{ p_2(H) }[/math] are surjective. Then one can apply Goursat's lemma to [math]\displaystyle{ H \leq p_1(H)\times p_2(H) }[/math].

To motivate the proof, consider the slice [math]\displaystyle{ S = \{g\} \times G' }[/math] in [math]\displaystyle{ G \times G' }[/math], for any arbitrary [math]\displaystyle{ g \in G }[/math]. By the surjectivity of the projection map to [math]\displaystyle{ G }[/math], this has a non trivial intersection with [math]\displaystyle{ H }[/math]. Then essentially, this intersection represents exactly one particular coset of [math]\displaystyle{ N' }[/math]. Indeed, if we have elements [math]\displaystyle{ (g,a), (g,b) \in S \cap H }[/math] with [math]\displaystyle{ a \in pN' \subset G' }[/math] and [math]\displaystyle{ b \in qN' \subset G' }[/math], then [math]\displaystyle{ H }[/math] being a group, we get that [math]\displaystyle{ (e, ab^{-1}) \in H }[/math], and hence, [math]\displaystyle{ (e, ab^{-1}) \in N' }[/math]. It follows that [math]\displaystyle{ (g,a) }[/math] and [math]\displaystyle{ (g,b) }[/math] lie in the same coset of [math]\displaystyle{ N' }[/math]. Thus the intersection of [math]\displaystyle{ H }[/math] with every "horizontal" slice isomorphic to [math]\displaystyle{ G' \in G\times G' }[/math] is exactly one particular coset of [math]\displaystyle{ N' }[/math] in [math]\displaystyle{ G' }[/math]. By an identical argument, the intersection of [math]\displaystyle{ H }[/math] with every "vertical" slice isomorphic to [math]\displaystyle{ G \in G\times G' }[/math] is exactly one particular coset of [math]\displaystyle{ N }[/math] in [math]\displaystyle{ G }[/math].

All the cosets of [math]\displaystyle{ N,N' }[/math] are present in the group [math]\displaystyle{ H }[/math], 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, [math]\displaystyle{ N }[/math] and [math]\displaystyle{ N' }[/math] are shown to be normal in [math]\displaystyle{ G \times \{e'\} }[/math] and [math]\displaystyle{ \{e\} \times G' }[/math], respectively. It is in this sense that [math]\displaystyle{ N }[/math] and [math]\displaystyle{ N' }[/math] can be identified as normal in G and G', respectively.

Since [math]\displaystyle{ p_2 }[/math] is a homomorphism, its kernel N is normal in H. Moreover, given [math]\displaystyle{ g \in G }[/math], there exists [math]\displaystyle{ h=(g,g') \in H }[/math], since [math]\displaystyle{ p_1 }[/math] is surjective. Therefore, [math]\displaystyle{ p_1(N) }[/math] is normal in G, viz:

[math]\displaystyle{ gp_1(N) = p_1(h)p_1(N) = p_1(hN) = p_1(Nh) = p_1(N)g }[/math].

It follows that [math]\displaystyle{ N }[/math] is normal in [math]\displaystyle{ G \times \{e'\} }[/math] since

[math]\displaystyle{ (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') }[/math].

The proof that [math]\displaystyle{ N' }[/math] is normal in [math]\displaystyle{ \{e\} \times G' }[/math] proceeds in a similar manner.

Given the identification of [math]\displaystyle{ G }[/math] with [math]\displaystyle{ G \times \{e'\} }[/math], we can write [math]\displaystyle{ G/N }[/math] and [math]\displaystyle{ gN }[/math] instead of [math]\displaystyle{ (G \times \{e'\})/N }[/math] and [math]\displaystyle{ (g,e')N }[/math], [math]\displaystyle{ g \in G }[/math]. Similarly, we can write [math]\displaystyle{ G'/N' }[/math] and [math]\displaystyle{ g'N' }[/math], [math]\displaystyle{ g' \in G' }[/math].

On to the proof. Consider the map [math]\displaystyle{ H \to G/N \times G'/N' }[/math] defined by [math]\displaystyle{ (g,g') \mapsto (gN, g'N') }[/math]. The image of [math]\displaystyle{ H }[/math] under this map is [math]\displaystyle{ \{(gN,g'N') \mid (g,g') \in H \} }[/math]. Since [math]\displaystyle{ H \to G/N }[/math] is surjective, this relation is the graph of a well-defined function [math]\displaystyle{ G/N \to G'/N' }[/math] provided [math]\displaystyle{ g_1N = g_2N \implies g_1'N' = g_2'N' }[/math] for every [math]\displaystyle{ (g_1,g_1'),(g_2,g_2') \in H }[/math], essentially an application of the vertical line test.

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

Furthermore, for every [math]\displaystyle{ (g_1,g_1'),(g_2,g_2')\in H }[/math] we have [math]\displaystyle{ (g_1g_2,g_1'g_2')\in H }[/math]. It follows that this function is a group homomorphism.

By symmetry, [math]\displaystyle{ \{(g'N',gN) \mid (g,g') \in H \} }[/math] is the graph of a well-defined homomorphism [math]\displaystyle{ G'/N' \to G/N }[/math]. 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
• J. Lambek (1996). "The Butterfly and the Serpent". Logic and Algebra. CRC Press. pp. 161–180. ISBN 978-0-8247-9606-8. 
• 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.