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 lemma also implies the snake lemma.
Goursat's lemma for groups can be stated as follows.
An immediate consequence of this is that the subdirect product of two groups can be described as a fiber product and vice versa.
Then one can apply Goursat's lemma to
To motivate the proof, consider the slice
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
lie in the same coset of
with every "horizontal" slice isomorphic to
By an identical argument, the intersection of
with every "vertical" slice isomorphic to
, 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.
are shown to be normal in
can be identified as normal in G and G', respectively.
is a homomorphism, its kernel N is normal in H. Moreover, given
proceeds in a similar manner.
Similarly, we can write
{\displaystyle (g,g')\mapsto (gN,g'N')}
{\displaystyle \{(gN,g'N')\mid (g,g')\in H\}}
is surjective, this relation is the graph of a well-defined function
, essentially an application of the vertical line test.
It follows that this function is a group homomorphism.
{\displaystyle \{(g'N',gN)\mid (g,g')\in H\}}
is the graph of a well-defined homomorphism
These two homomorphisms are clearly inverse to each other and thus are indeed isomorphisms.
As a consequence of Goursat's theorem, one can derive a very general version on the Jordan–Hölder–Schreier theorem in Goursat varieties.