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.
Groups
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
. One then obtains a bijection between: - Subgroups of
which project onto both factors, - Triples
with
normal in
,
normal in
and
isomorphism of
onto
.
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 have elements
with
and
, then
being a group, we get that
, and hence,
. It follows that
and
lie in the same coset of
. 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
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
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". 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.
- 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.