Goursat's lemma

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

Goursat's lemma for groups can be stated as follows.

Let

G

G'

be groups, and let

H

be a subgroup of

G\times G'

such that the two projections

p_{1}:H\rightarrow G

and

p_{2}:H\rightarrow G'

are surjective (i.e.,

H

is a subdirect product of

G

and

G'

). Let

N

be the kernel of

p_{2}

and

N'

the kernel of

p_{1}

. One can identify

N

as a normal subgroup of

G

, and

N'

as a normal subgroup of

G'

. Then the image of

H

G/N\times G'/N'

is the graph of an isomorphism

G/N\approx G'/N'

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 $S={g}\times G'$ in $G\times G'$ , for any arbitrary ${g}\in G$ . By the surjectivity of the projection map to $G$ , this has a non trivial intersection with $H$ . Then essentially, this intersection represents exactly one particular coset of $G'$ . Indeed, if we had distinct elements $(g,a),(g,b)\in S$ with $a\in pN'\subset G'$ and $a\in qN'\subset G'$ , then $H$ being a group, we get that $(e,ab^{-1})\in H$ , and hence, $(e,ab^{-1})\in N'$ . But this a contradiction, as $a,b$ belong to distinct cosets of $N'$ , and thus $ab^{-1}N'\neq N'$ , and thus the element $(e,ab^{-1})\in N'$ cannot belong to the kernel $N'$ of the projection map from $H$ to $G$ . Thus the intersection of $H$ with every "horizontal" slice isomorphic to $G'\in G\times G'$ is exactly one particular coset of $N'$ in $G'$ . By an identical argument, the intersection of $H$ with every "vertical" slice isomorphic to $G\in G\times G'$ is exactly one particular coset of $N$ in $G$ .

All the cosets of $G,G'$ are present in the group $H$ , 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, $N$ and $N'$ are shown to be normal in $G\times \{e'\}$ and $\{e\}\times G'$ , respectively. It is in this sense that $N$ and $N'$ can be identified as normal in G and G', respectively.

Since $p_{2}$ is a homomorphism, its kernel N is normal in H. Moreover, given $g\in G$ , there exists $h=(g,g')\in H$ , since $p_{1}$ is surjective. Therefore, $p_{1}(N)$ is normal in G, viz:

gp_{1}(N)=p_{1}(h)p_{1}(N)=p_{1}(hN)=p_{1}(Nh)=p_{1}(N)g

It follows that $N$ is normal in $G\times \{e'\}$ since

(g,e')N=(g,e')(p_{1}(N)\times \{e'\})=gp_{1}(N)\times \{e'\}=p_{1}(N)g\times \{e'\}=(p_{1}(N)\times \{e'\})(g,e')=N(g,e')

The proof that $N'$ is normal in $\{e\}\times G'$ proceeds in a similar manner.

Given the identification of $G$ with $G\times \{e'\}$ , we can write $G/N$ and $gN$ instead of $(G\times \{e'\})/N$ and $(g,e')N$ , $g\in G$ . Similarly, we can write $G'/N'$ and $g'N'$ , $g'\in G'$ .

On to the proof. Consider the map $H\rightarrow G/N\times G'/N'$ defined by $(g,g')\mapsto (gN,g'N')$ . The image of $H$ under this map is $\{(gN,g'N')|(g,g')\in H\}$ . Since $H\rightarrow G/N$ is surjective, this relation is the graph of a well-defined function $G/N\rightarrow G'/N'$ provided $g_{1}N=g_{2}N\Rightarrow g_{1}'N'=g_{2}'N'$ for every $(g_{1},g_{1}'),(g_{2},g_{2}')\in H$ , essentially an application of the vertical line test.

Since $g_{1}N=g_{2}N$ (more properly, $(g_{1},e')N=(g_{2},e')N$ ), we have $(g_{2}^{-1}g_{1},e')\in N\subset H$ . Thus $(e,g_{2}'^{-1}g_{1}')=(g_{2},g_{2}')^{-1}(g_{1},g_{1}')(g_{2}^{-1}g_{1},e')^{-1}\in H$ , whence $(e,g_{2}'^{-1}g_{1}')\in N'$ , that is, $g_{1}'N'=g_{2}'N'$ .

Furthermore, for every $(g_{1},g_{1}'),(g_{2},g_{2}')\in H$ we have $(g_{1}g_{2},g_{1}'g_{2}')\in H$ . It follows that this function is a group homomorphism.

By symmetry, $\{(g'N',gN)|(g,g')\in H\}$ is the graph of a well-defined homomorphism $G'/N'\rightarrow G/N$ . 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. 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.

This article is issued from Wikipedia - version of the 10/8/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.