Goursat's lemma

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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.

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 had distinct elements with and , then being a group, we get that , and hence, . But this a contradiction, as belong to distinct cosets of , and thus , and thus the element cannot belong to the kernel of the projection map from to . 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[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]

  • 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". In Aldo Ursini, Paulo Agliano. 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.