Goursat's lemma

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.

GroupsEdit

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.

Notice that if   is any subgroup of   (the projections   and   need not be surjective), then the projections from   onto   and   are surjective. Then one can apply Goursat's lemma to  .

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.

ProofEdit

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 varietiesEdit

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

ReferencesEdit

  • 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 (eds.). 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.