Goursat's lemma

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Not to be confused with Goursat's integral lemma from complex analysis.

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[edit]

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 .

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

Proof[edit]

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[edit]

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

References[edit]