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.

Dataset: 1312.1485v2
 SourceAvailable from: László Tóth[Show abstract] [Hide abstract]
ABSTRACT: We discuss properties of the subgroups of the group Z_m × Z_n , where m and n are arbitrary positive integers. Simple formulae for the total number of subgroups and the number of subgroups of a given order are deduced. The cyclic subgroups and subgroups of a given exponent are also considered.Journal of Numbers. 11/2012;  SourceAvailable from: László Tóth[Show abstract] [Hide abstract]
ABSTRACT: Using Goursat's lemma for groups, a simple representation and the invariant factor decompositions of the subgroups of the group Z_m x Z_n are deduced, where m and n are arbitrary positive integers. As consequences, explicit formulas for the total number of subgroups, the number of subgroups with a given invariant factor decomposition, and the number of subgroups of a given order are obtained.Tatra Mt. Math. Publ. 09/2014;
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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 1112 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
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There are a number of interesting possibilities for generalizing this useful
lemma. The first is to subgroups of a semidirect 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 LubkinFreyd
HeronMitchell (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
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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
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Finally, we have
Ker(θ1) = {a ∈ G1θ1(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(jS) := {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(12,3).
Note that if T ⊆ S, then G(jS) ✁ G(jT).
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
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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 7tuples
{G1,G2,G(21),θ1,G3,G(31,2),θ2},
where G1⊆ A, G(21) ✁G2⊆ B, G(31,2) ✁G3⊆ C, θ1: G1։ G2/G(21),
and θ2: Λ ։ G3/G(31,2) are surjective homomorphisms with
Λ = Γ2({G1,G2,G(21),θ1}) ⊆ A × B
(cf. Theorem 2.2 for the definition of Γ2).
Proof.
If G is a subgroup of A × B × C, define the 7tuple
Q3(G) = {G1,G2,G(21),θ1,G3,G(31,2),θ2}
where G1, G2, G(21), G3, and G(31,2) 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 7tuple
Q = {G1,G2,G(21),θ1,G3,G(31,2),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(31,2) 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 4tuple Q2(G)
for n = 2 and 7tuple 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
Page 6
4Application to cyclic subgroups
There are many potentially interesting applications of the (generalized)
Goursat lemma within group theory. For example, here are four immediate
applications.
4.1 Proposition Let G be a subgroup of A1× ··· × An.
(a) The subgroup G is finite if and only if G1,··· ,Gnare all finite.
(b) The subgroup G is abelian if and only if G1,··· ,Gnare all abelian.
(c) The subgroup G is pgroup if and only if G1,··· ,Gnare all pgroups.
(d) The subgroup G is a subdirect product if and only if θj= 0 for 1 ≤ j ≤
n − 1.
It is interesting to note that Hattori, in [Hat61] or [HZ09], first deter
mined the finite subgroups of the Lie group S3(which is isomorphic to
SU(2)≈ Sp(1)≈ Spin(3) as a Lie group). He then applied Goursat’s lemma
and Proposition 4.1(a) to determine all finite subgroups of S3× S3. Using
the results of Section 3 we could now, for example, find all finite subgroups
of S3× ···
n× S3. It is also interesting that, in fact, the papers of Goursat
[Gou89] and Hattori [Hat61] or [HZ09] study closely related questions.
The next application, that of determining the cyclic subgroups of A×B,
will involve more substantial use of Goursat’s lemma . We shall henceforth
use additive notation since G1,G1,G2,G2will be abelian.
4.2 Theorem Let G be a subgroup of A × B.
(a) The subgroup G is finite cyclic if and only if G1,G2are both finite cyclic
and G1,G2have coprime order. Furthermore, G = lcm(G1,G2).
(b) The subgroup G is infinite cyclic if and only if either G1 is infinite
cyclic, G2 is finite cyclic, and G2 = {0}, or G2 is infinite cyclic, G1
is finite cyclic, and G1 = {0}, or both G1,G2 are infinite cyclic with
G1= G2= {0}.
Proof.
a cyclic group, must be cyclic, hence their respective subgroups G1,G2are
also cyclic.
(a) Suppose G is finite cyclic, then it is generated by an element (α,β),
whence G1 is cyclic and generated by α, G2 cyclic and generated by β.
In either case (a) or (b), G1or G2being the homomorphic image of
6
Page 7
Let the respective orders of G1,G2 (i.e.
gcd(m,n). Also write m = m1d, n = n1d. Then m1,n1are coprime, and
there exist x,y ∈ Z with xm1+ yn1 = 1, or equivalently xm + yn = d.
Now n(α,β) = (nα,0) ∈ G implies nα ∈ G1. Also mα = 0 ∈ G1. Hence
dα = (xm + yn)α = x(mα) + y(nα) ∈ G1. It follows that cα / ∈ G1 if
0 < c < d. For, if cα ∈ G1, then (cα,0) ∈ G. Hence (cα,0) = z(α,β) for
some integer z. Therefore zβ = 0 = (cz)α, whence nz and m(cz). Since
d divides m and n, we have d divides both z,cz. As a result, dc, which
is a contradiction. Thus we conclude that G1is the cyclic subgroup of G1,
generated by dα and having order m/d = m1. Similarly, G2is generated by
dβ and has order n1, so the orders of G1and G2are coprime.
Conversely, suppose G1 = m1 is coprime to G2 = n1, and set d =
G1/G1 = G2/G2, m = m1d,n = n1d. Then d = gcd(m,n). Also
G1will be cyclic of order m, G2cyclic of order n. Using the isomorphism
θ: G1/G1→ G2/G2, choose a generator α of G1and let θ[α] = [β]. Then [α]
and [β] are both elements of of order d, whence β has order dG2 = dn1= n
and generates G2. Also, by Goursat’s lemma, γ = (α,β) ∈ G. Finally, the
order of γ is lcm(o(α),o(β))=lcm(m,n) = (mn/d) = dm1n1. Further, again
using Goursat’s lemma, G = GθG1G2 = dm1n1. Thus G is cyclic of
this order and generated by (α,β).
(b) Since G is infinite and G ⊆ G1× G2, at least one of G1,G2must be
infinite cyclic. Without loss of generality, suppose G1≈ Z. Now suppose
(α,β) generates the cyclic group G, then α generates G1, and β generates
G2. We claim that G2= {0}. For, if y ∈ G2then y = rβ for some integer
r, whence (0,y) = r(0,β) ∈ G. This implies (0,y) = k(α,β) = (kα,kβ)
for some integer k. Therefore kα = 0, whence k = 0 and y = kβ = 0.
Hence G2= {0}. We now consider separately the cases G2infinite and G2
finite (the case G1finite and G2≈ Z is symmetric to the latter, so can be
omitted).
Suppose first G ≈ Z with G2≈ Z. Then the argument in the previous
paragraph now also implies G1= {0}. Conversely, suppose G1≈ G2≈ Z
and G1= G2= {0}. Then the isomorphisms (cf. Remark 2.3)
of α,β) be m,n, and set d =
G/(G1× G2) ≈ G1/G1
θ− → G2/G2
reduce to G ≈ G1≈ G2≈ Z.
Secondly, for the remaining case, suppose G ≈ Z,G1≈ Z as before and
now G2≈ Znis cyclic of order n, n ≥ 2. Then n(α,β) = (nα,0) implies
nα ∈ G1 and clearly iα / ∈ G1 if i < n. Thus G1 ≈ nZ, and as before
G2= {0}.
7
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Conversely, suppose G1≈ Z,G1≈ nZ,G2≈ Znand G2= {0}. In this
case we have the isomorphism θ: G1/G1→ G2/G2≈ Zn. Let α ∈ G1with
[α] generating G1/G1. Then θ[α] = [β] = β generates G2/G2= G2≈ Zn.
We claim that G is generated by the single element (α,β), and thus is
infinite cyclic. To see this, let (x,y) ∈ G ⊆ G1× G2, so x = jα,y = kβ
for some integers j,k.Furthermore (x,y) ∈ G implies θ[x] = [y] = y,
which gives kβ = y = θ[jα] = j(θ[α]) = jβ. Then j ≡ k(mod n), so
j(α,β) = (jα,jβ) = (jα,kβ) = (x,y).
One could ask the same question for most other properties of groups, e.g.
normality, nilpotency, and solvabability to name a few. This will undoubt
edly lead to interesting results in many cases, which are beyond the scope
of the present investigation. We conclude this note with the generalization
of Theorem 4.2 to cyclic subgroups of higher direct products.
?
4.3 Theorem Let G be a subgroup of A × B × C.
(a) The subgroup G is finite cyclic if and only if G1,G2,G3are finite cyclic
and each of the pairs of integers (G(12),G(21)), (G(13),G(31)),
(G(23),G(32)) is coprime. In this case one also has
G = lcm(G1,G2,G3).
(b) The subgroup G is infinite cyclic if and only if one of the following three
cases (up to obvious permutation of indices) occur :
(i)
G1≈ Z, G2and G3are finite cyclic, G(21) = G(31) = {0},
and G(23),G(32) are coprime.
(ii)
G1 ≈ G2 ≈ Z, G3 finite cyclic, and G(21) = G(31) =
G(12) = G(32) = {0}.
(iii)Gi≈ Z for i = 1,2,3 and G(ij) = 0 for 1 ≤ i ?= j ≤ 3.
Proof.
part (a) of Theorem 4.2 to G12gives us G1,G2finite cyclic with G(12)
coprime to G(21). The other conditions follow by symmetry.
Conversely, suppose G1,G2,G3 are all finite cyclic with respective or
ders m,n,p, and that the three coprimality conditions hold.
Theorem 4.2(a) two times we obtain that G12 and G12 are both finite
cyclic with respective orders lcm(m,n), lcm(G(13),G(23)).
apply Theorem 3.3, which tells us that G is determined by the surjec
tion θ2: G12։ G3/G(31,2), or equivalently by the induced isomorphism
φ: G12/G12
− → G3/G(31,2). A third application of Theorem 4.2(a) now
(a) If G is finite cyclic, then so is G12= π12(G) ⊆ A×B. Applying
Applying
We now
≈
8
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tells us that G will be cyclic if G12 and G(31,2) are coprime.
G12 = lcm(G(13),G(23)), and G(31,2) is a divisor of G(31) which is
coprime to G(13). Thus G(31,2) is coprime to G(13), and similarly is
coprime to G(23), so also coprime to their least common multiple G12.
(b) The three cases when G is infinite cyclic all follow from Theorem
4.2(b) in obvious ways, namely in (b)(i) we use A × B × C ≈ A × (B × C),
in (b)(ii) and (b)(iii) we use A×B×C ≈ (A×B)×C. We omit the details.
?
The generalization of this theorem to n ≥ 3 is now clear.
But
References
[AC09] D. D. Anderson and V. Camillo, Subgroups of direct products of
groups, ideals and subrings of direct products of rings, and Goursat’s
lemma, Rings, modules and representations, Contemp. Math., vol.
480, Amer. Math. Soc., Providence, RI, 2009, pp. 1–12. MR 2508141
(2010h:20063)
[AEM09] C. Arroyo, S. Eggleston, B. MacGregor, Applications and gener
alizations of Goursat’s lemma, http://www.slideshare.net/dadirac
/goursatslemmapresentation2411944 (2009).
[Dic69] S. E. Dickson, On algebras of finite representation type, Trans.
Amer. Math. Soc. 135 (1969), 127–141. MR 0237558 (38 #5839)
[FL10] J. F.Farrill and S. Lack, For which categories does one have a Gour
sat lemma?, http://mathoverflow.net/questions/46700/for
whichcategoriesdoesonehaveagoursatlemma (2010).
[Gou89]´E. Goursat, Sur les substitutions orthogonales et les divisions
r´ eguli` eres de l’espace, Ann. Sci.´Ecole Norm. Sup. (3) 6 (1889), 9–
102. MR 1508819
[Hat61] A. Hattori, On 3dimensional elliptic space forms, S¯ ugaku 12
(1960/1961), 164–167. MR 0139119 (25 #2558)
[HZ09] L. Martins, S. Massago, M. Mimura, A. Hattori and P. Zvengrowski,
Threedimensional spherical space forms, Group actions and homoge
neous spaces, Proceedings of the International Conference Bratislava
Topology Symposium, 2009, pp. 29–42.
9
Page 10
[Lam58] J. Lambek, Goursat’s theorem and the Zassenhaus lemma, Canad.
J. Math. 10 (1958), 45–56. MR 0098138 (20 #4600)
[Lam76]
, Lectures on Rings and Modules, second ed., Chelsea Pub
lishing Co., New York, 1976. MR 0419493 (54 #7514)
[Lan02] S. Lang, Algebra, revised third ed., AddisonWesley Publishing
Company Advanced Book Program, Reading, MA, 2002. MR 783636
(86j:00003)
[Mac71] S. MacLane, Categories for the Working Mathematician, Springer
Verlag, New York, 1971, Graduate Texts in Mathematics, Vol. 5. MR
0354798 (50 #7275)
[Pet09] J. Petrillo, Goursat’s other theorem, College Math. J. 40, NO. 2
(2009), 119–124.
[Pet11], Counting subgroups in a direct product of finite cyclic
groups, College Math. J. 42, NO. 3 (2011), 215–222.
[Rib76] K. A. Ribet, Galois action on division points of abelian varieties
with real multiplications, Amer. J. Math. 98 (1976), no. 3, 751–804.
MR 0457455 (56 #15660)
[Sch94] R. Schmidt, Subgroup Lattices of Groups, de Gruyter Expositions
in Mathematics, vol. 14, Walter de Gruyter & Co., Berlin, 1994. MR
1292462 (95m:20028)
[Use91] V. M. Usenko, Subgroups of semidirect products, Ukrain. Mat. Zh.
43 (1991), no. 78, 1048–1055. MR 1148867 (92k:20045)
Kristine Bauer
Department of Mathematics and Statistics
University of Calgary
Calgary, Alberta, Canada T2N 1N4
email: bauerk@ucalgary.ca
Debasis Sen
Department of Mathematics and Statistics
University of Calgary
Calgary, Alberta, Canada T2N 1N4
email: sen.deba@gmail.com
Peter Zvengrowski
Department of Mathematics and Statistics
10
Page 11
University of Calgary
Calgary, Alberta, Canada T2N 1N4
email: zvengrow@ucalgary.ca
11
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