A Generalized Goursat Lemma
ABSTRACT In this note the usual Goursat lemma, which describes subgroups of the direct
product of two groups, is generalized to describing subgroups of a direct
product \ $A_1\times A_2 \times...\times A_n$ \ of a finite number of groups.
Other possible generalizations are discussed and an application to cyclic
subgroups is given.
- Citations (1)
- Cited In (0)
- MR 0098138 (20 #4600) [Lam76] , Lectures on Rings and Modules Chelsea Pub-lishing Co. MR 0419493 (54 #7514) [Lan02] S. Lang, Algebra, revised third ed. J Math . 45-5600003.
Page 1
arXiv:1109.0024v2 [math.GR] 19 Sep 2011
A GENERALIZED GOURSAT LEMMA
K. Bauer, D. Sen, P. Zvengrowski
Abstract
In this note the usual Goursat lemma, which describes subgroups
of the direct product of two groups, is generalized to describing sub-
groups of a direct product A1× A2× ··· × An of a finite number of
groups. Other possible generalizations are discussed and an application
to cyclic subgroups is given.
1 Introduction
In Sections 11-12 of a paper written in 1889 [Gou89], the famed French math-
ematician´Edouard Goursat developed what is now called Goursat’s lemma
(also called Goursat’s theorem or Goursat’s other theorem), for characteriz-
ing the subgroups of the direct product A×B of two groups A,B. It seems
to have been first attributed to Goursat by J. Lambek in [Lam58, Lam76],
who in turn attributes H.S.M. Coxeter for bringing this to his attention.
The lemma is elementary and a natural question to consider, for example
it appears as Exercise 5, p. 75, in Lang’s Algebra [Lan02]. It has also been
the subject of recent expository articles [Pet09, Pet11] in an undergraduate
mathematics journal. It is possible that other authors discovered the lemma
independently without knowing the original reference. Indeed, one such ex-
ample occurs in 1961 in a paper of A. Hattori, ([Hat61], Section 2.3), now
translated into English [HZ09] from the original Japanese.
Other sources that mention Goursat’s lemma include papers of S. Dick-
son [Dic69] in 1969, K. Ribet [Rib76] in 1976, a book by R. Schmidt ([Sch94],
Chapter 1.6), a paper of D. Anderson and V. Camillo in 2009 [AC09], and
several internet sites such as [FL10, AEM09].
12010 Mathematics Subject Classification : 20, 18.
Keywords and phrases : Goursat’s Lemma
2The third named author was supported during this work by a Discovery Grant from
the Natural Sciences and Engineering Research Council of Canada.
1
Page 2
There are a number of interesting possibilities for generalizing this useful
lemma. The first is to subgroups of a semi-direct product, and this is stud-
ied in [Use91]. The second is to other categories besides groups. Indeed,
it is proved for modules in [Lam76], and this implies that it will hold in
any abelian category by applying the embedding theorems of Lubkin-Freyd-
Heron-Mitchell (cf. [Mac71], p. 205). It is proved for rings in [AC09]. The
most general category in which one can hope to have a Goursat lemma is
likely an exact Mal’cev category, cf. [FL10], although to the best of our
knowledge a a proof of this fact is yet to be given.
In this note we examine another generalization, to the direct product of
a finite number of groups. While this sounds at first glance to be a triviality
since we can write A×B×C ≈ (A×B)×C, unexpected complications arise
as noted by Arroyo et al. in [AEM09]. The complications are overcome by
introducing an asymmetric version of the lemma in Section 2, which enables
us to solve the general case in Section 3. Applications within group theory,
in particular to cyclic subgroups of a direct product A1×···×An, are given
in Section 4.
2 An asymmetric version of Goursat’s lemma
For convenience, we start by stating the usual version of Goursat’s lemma.
Let A,B be groups and G ⊆ A × B be a subgroup. The neutral element of
each group A and B, with slight abuse of notation, will be written ‘e’. Let
π1: A × B → A, π2: A × B → B be the natural projections and ı1: A →
A × B, ı2: B → A × B be the usual inclusions.
2.1 Theorem (Goursat’s lemma) There is a bijective correspondence be-
tween subgroups G of A × B and quintuples {G1,G1,G2,G2,θ}, where
G1✁ G1⊆ A, G2✁ G2⊆ B, and θ: G1/G1
≈
− → G2/G2is an isomorphism.
As mentioned above, the proof is elementary and given as an exercise in
[Lan02], it can also be found in [AC09, Hat61]. The basic idea of the proof
is as follows. Suppose that G is a subgroup of A × B. Write G1= π1(G) =
{a ∈ A|(a,b) ∈ G for some b ∈ B}, G1 = ı−1
similarly for G2 and G2. It is easily seen that G1✁ G1, G2✁ G2. The
isomorphism θ: G1/G1
− → G2/G2is given by θ[a] = [b], where (a,b) ∈ G
and [a] = G1a, [b] = G2b are the respective cosets of a and b in G1/G1,
G2/G2(again with slight abuse of notation). Thus G determines a quintuple
Q′
2(G) = {G1,G1,G2,G2,θ}.
1(G) = {a ∈ A|(a,e) ∈ G},
≈
2
Page 3
Conversely, given a quintuple Q′= {G1,G1,G2,G2,θ}, let Γ′
the subgroup p−1(Gθ), where Gθ⊆ G1/G1× G2/G2is the graph of θ and
p: G1× G2→ G1/G1× G2/G2is the natural surjection. The functions Q′
and Γ′
2are inverse to each other.
We now state an equivalent asymmetric version of the lemma, which is
in effect just a minor variation of Theorem 2.1 but has the advantage that
it generalizes easily to higher direct products, as we shall see in Section 3.
2(Q′) be
2
2.2 Theorem (Asymmetric version of Goursat’s lemma) There is a
bijective correspondence between subgroups G of A × B and quadruples
{G1,G2,G2,θ1}, where G1⊆ A, G2✁ G2⊆ B are arbitrary subgroups of A
and B, and θ1: G1։ G2/G2is a surjective homomorphism.
Proof.
A × B we define Q2(G) to be the quadruple
The proof is similar to that of Theorem 2.1. For any subgroup G of
{G1,G2,G2,θ1},
where G1, G2and G2are the first, third and fourth coordinates of Q′
(cf. Theorem 2.1). The surjection θ1is given by θ1(a) = [b] for a ∈ G1and
(a,b) ∈ G for some b ∈ G2.
Conversely, for an arbitrary quadruple Q = {G1,G2,G2,θ1}, with G1⊆
A, G2✁ G2⊆ B, and θ1: G1։ G2/G2a surjective homomorphism define
2(G)
Γ2(Q) := p−1(Gθ1),
where Gθ1⊆ G1× G2/G2is the graph of θ1and p: G1× G2→ G1× G2/G2
is the natural surjection. The functions Q2and Γ2are inverse to each other.
?
The equivalence of Theorem 2.1 and Theorem 2.2 is easily seen. It need
only be pointed out that θ determines the surjection θ1as the composition
G1։ G1/G1
θ
≈G2/G2,
− →
while θ1determines θ via the first isomorphism theorem, specifically
G1
G2/G2
????????
???
???
????
G1/Ker(θ1)
??
θ1
??????????????
≈
θ
.
3
Page 4
Finally, we have
Ker(θ1) = {a ∈ G1|θ1(a) = [b],with b ∈ G2}
= {a ∈ G1|(a,b) ∈ G, (e,b) ∈ G}
= {a ∈ G1|(a,e) ∈ G}
= G1.
2.3 Remark It is often useful to note the additional fact that (G1×G2)✁G
and G/(G1× G2) ≈ G1/G1≈ G2/G2[Hat61], [HZ09].
3 The generalized Goursat lemma
As mentioned in the Introduction, generalizing the Goursat lemma from
n = 2 to n ≥ 2 seems to create unexpected complications. However, using
the asymmetric version Theorem 2.2, the generalization to finite n ≥ 2
becomes routine. We will next state the result for n = 3, after introducing
some convenient notation for any subgroup G ⊆ A1× ··· × An.
3.1 Definition Let S ⊂ {1,2,··· ,n} = n, and j ∈ n ? S. Then
G(j|S) := {xj∈ Aj|(x1,··· ,xj,··· ,xn) ∈ G
for some x1,··· ,xj−1,xj+1,··· ,xnwith xi= e if i ∈ S}.
For example,
G(1|∅) = π1(G), G(1|{2,3,··· ,n − 1}) = {x1∈ A1|(x1,e,··· ,e)} ∈ G}.
These correspond to the notation used in Section 2, when n = 2, via
G(1|∅) = G1, G(1|{2}) = G1, G(2|∅) = G2, G(2|{1}) = G2. For conve-
nience, we shall usually omit the brackets {}, e.g. G(1|{2,3}) = G(1|2,3).
Note that if T ⊆ S, then G(j|S) ✁ G(j|T).
3.2 Definition With S as above, we denote by πSthe standard projection
A1× ··· × An։?
··· × An. Then we set
j∈SAjand ıSthe standard inclusion?
j∈SAj֒→ A1×
GS:= πS(G) ⊆
?
j∈S
Aj,GS:= ı−1
S(G) ⊆
?
j∈S
Aj.
4
Page 5
For example, if n = 3, A1= A, A2= B, A3= C, and S = {1,3}, then
GS= {(a,c)|(a,b,c) ∈ G for some b ∈ B} and GS= {(a,c)|(a,e,c) ∈ G}. In
case S = {j}, GS= G{j}is written Gj, which is consistent with the notation
in Section 2, and similarly GS = G{j}= Gj = {aj ∈ Aj|(a1,...,an) ∈
G with ai = e,i ?= j}. One has GS✁ GS ⊆ A × C. We now state the
generalized Goursat lemma for n = 3, the extension for n ≥ 3 should be
clear (cf. Remark 3.4).
3.3 Theorem (Goursat’s lemma for n = 3) There is a bijective corre-
spondence between the subgroups of A × B × C and 7-tuples
{G1,G2,G(2|1),θ1,G3,G(3|1,2),θ2},
where G1⊆ A, G(2|1) ✁G2⊆ B, G(3|1,2) ✁G3⊆ C, θ1: G1։ G2/G(2|1),
and θ2: Λ ։ G3/G(3|1,2) are surjective homomorphisms with
Λ = Γ2({G1,G2,G(2|1),θ1}) ⊆ A × B
(cf. Theorem 2.2 for the definition of Γ2).
Proof.
If G is a subgroup of A × B × C, define the 7-tuple
Q3(G) = {G1,G2,G(2|1),θ1,G3,G(3|1,2),θ2}
where G1, G2, G(2|1), G3, and G(3|1,2) are the groups of Definitions 3.1
and 3.2. Further, for a ∈ G1we have θ1(a) = [b], where (a,b,c) ∈ G for some
b ∈ B,c ∈ C , and for (a,b) ∈ Λ, we have θ2(a,b) = [c] where (a,b,c) ∈ G
for some c ∈ C.
Conversely, given any 7-tuple
Q = {G1,G2,G(2|1),θ1,G3,G(3|1,2),θ2},
satisfying the hypotheses of Theorem 3.3, a subgroup Γ3(Q) ⊆ A × B × C
is determined by Γ3(Q) = p−1(Gθ2), where Gθ2⊆ Λ×G3⊆ A×B ×C is the
graph of θ2and p: Λ × G3→ Λ × G3/G(3|1,2) is the natural surjection.
The fact that Γ3 and Q3 are inverse to each other is immediate from
the case n = 2. We simply apply Theorem 2.2 recursively to A × B × C =
(A × B) × C, first to G1⊆ A, G2⊆ B to obtain Λ = G{1,2}⊆ A × B, and
then to G{1,2}⊆ A × B, G3⊆ C to obtain G.
?
3.4 Remark To extend to n ≥ 4, we simply note that the 4-tuple Q2(G)
for n = 2 and 7-tuple Q3(G) for n = 3 will be replaced by a suitable (3n−2)-
tuple Qn(G), noting that 3n − 2 = 4 + 3(n − 2).
5