A \textit{$3$-net} of order $n$ is a finite incidence structure consisting of
points and three pairwise disjoint classes of lines, each of size $n$, such
that every point incident with two lines from distinct classes is incident with
exactly one line from each of the three classes. The current interest around
$3$-nets (embedded) in a projective plane $PG(2,K)$, defined over a field $K$
of characteristic $p$, arose from algebraic geometry. It is not difficult to
find $3$-nets in $PG(2,K)$ as far as $0<p\le n$. However, only a few infinite
families of $3$-nets in $PG(2,K)$ are known to exist whenever $p=0$, or $p>n$.
Under this condition, the known families are characterized as the only $3$-nets
in $PG(2,K)$ which can be coordinatized by a group. In this paper we deal with
$3$-nets in $PG(2,K)$ which can be coordinatized by a diassociative loop $G$
but not by a group. We prove two structural theorems on $G$. As a corollary, if
$G$ is commutative then every non-trivial element of $G$ has the same order,
and $G$ has exponent $2$ or $3$. We also discuss the existence problem for such
$3$-nets

Korchmáros, Gábor

Nagy, Gábor P.

English

arXiv

A 3-net of order n is a finite incidence structure consisting of points and three pairwise disjoint classes of lines, each of size n, such that every point incident with two lines from distinct classes is incident with exactly one line from each of the three classes. The current interest around 3-nets (embedded) in a projective plane PG(2, K), defined over a field K of characteristic p, arose from algebraic geometry; see Falk and Yuzvinsky (Compos Math 143: 1069-1088, 2007), Miguel and Buzunariz (Graphs Comb 25: 469-488, 2009), Pereira and Yuzvinsky (Adv Math 219: 672-688, 2008), Yuzvinsky (140: 1614-1624, 2004), and Yuzvinsky (137: 1641-1648, 2009). It is not difficult to find 3-nets in PG(2, K) as far as 0  n. Under this condition, the known families are characterized as the only 3-nets in PG(2, K) which can be coordinatized by a group; see Korchmaros et al. (J Algebr Comb 39: 939-966, 2014). In this paper we deal with 3-nets in PG(2, K) which can be coordinatized by a diassociative loop G but not by a group. We prove two structural theorems on G. As a corollary, if G is commutative then every non-trivial element of G has the same order, and G has exponent 2 or 3 where the exponent of a finite diassociative loop is the maximum of the orders of its elements. We also discuss the existence problem for such 3-nets

Korchmaros Gábor

Nagy Gábor Péter

SZTE Publicatio Repozitórium - SZTE - Repository of Publications

3-Nets realizing a diassociative loop in a projective plane

Gábor Korchmáros

Gábor P. Nagy

Crossref

Designs Codes and Cryptography

A 3-net of order n is a finite incidence structure consisting of points and three pairwise disjoint classes of lines, each of size n, such that every point incident with two lines from distinct classes is incident with exactly one line from each of the three classes. The current interest around 3-nets (embedded) in a projective plane (Formula presented.) , defined over a field (Formula presented.) of characteristic p, arose from algebraic geometry; see Falk and Yuzvinsky (Compos Math 143:1069–1088, 2007), Miguel and BuzunÃ¡riz (Graphs Comb 25:469–488, 2009), Pereira and Yuzvinsky (Adv Math 219:672–688, 2008), Yuzvinsky (140:1614–1624, 2004), and Yuzvinsky (137:1641–1648, 2009). It is not difficult to find 3-nets in (Formula presented.) as far as (Formula presented.). However, only a few infinite families of 3-nets in (Formula presented.) are known to exist whenever (Formula presented.) , or (Formula presented.). Under this condition, the known families are characterized as the only 3-nets in (Formula presented.) which can be coordinatized by a group; see KorchmÃ¡ros et al. (J Algebr Comb 39:939–966, 2014). In this paper we deal with 3-nets in (Formula presented.) which can be coordinatized by a diassociative loop G but not by a group. We prove two structural theorems on G. As a corollary, if G is commutative then every non-trivial element of G has the same order, and G has exponent 2 or 3 where the exponent of a finite diassociative loop is the maximum of the orders of its elements. We also discuss the existence problem for such 3-nets

KORCHMAROS, Gabor

Nagy, Gabor P.

Archivio della Ricerca - Università della Basilicata

Korchmaros, G

Nagy, GP

ELTE Digital Institutional Repository (EDIT)

arXiv:1603.00242v1  [math.GR]  1 Mar 20163-NETS REALIZING A DIASSOCIATIVE LOOP IN APROJECTIVE PLANEGÁBOR KORCHMÁROS AND GÁBOR P. NAGYAbstract. A 3-net of order n is a finite incidence structure consisting ofpoints and three pairwise disjoint classes of lines, each of size n, such that ev-ery point incident with two lines from distinct classes is incident with exactlyone line from each of the three classes. The current interest around 3-nets(embedded) in a projective plane PG(2,K), defined over a field K of character-istic p, arose from algebraic geometry; see [5, 12, 14, 17, 18]. It is not difficultto find 3-nets in PG(2,K) as far as 0 < p ≤ n. However, only a few infinitefamilies of 3-nets in PG(2,K) are known to exist whenever p = 0, or p > n.Under this condition, the known families are characterized as the only 3-netsin PG(2,K) which can be coordinatized by a group; see [10]. In this paperwe deal with 3-nets in PG(2,K) which can be coordinatized by a diassociativeloop G but not by a group. We prove two structural theorems on G. As acorollary, if G is commutative then every non-trivial element of G has the sameorder, and G has exponent 2 or 3. We also discuss the existence problem forsuch 3-nets.Keywords 3-net - projective plane - diassociative loop - Latin square - transversaldesignMathematics Subject Classification 51E99 20N051. IntroductionThe concept of a 3-net comes from classical differential geometry via the com-binatorial abstraction of the concept of a 3-web. Formally, a 3-net of order n isa finite incidence structure consisting of points and three pairwise disjoint classesof lines, each of size n, such that every point incident with two lines from distinctclasses is incident with exactly one line from each of the three classes. It is wellknown that every 3-net can be coordinatized by a loop. The set Q endowed witha binary operation “·” is a quasigroup, if for any a, b ∈ Q, the equations a · x = band y · a = b have unique solutions in Q. A quasigroup with a multiplicative unitelement is called a loop. For a general reference on nets, loops and quasigroups seefor instance [1, 4].In this paper we deal with 3-nets (embedded) in PG(2,K), the projective planeover a field K of characteristic p ≥ 0. Such a 3-net, with line classes A,B, C andcoordinatizing loop G = (G, ·), is equivalently defined by a triple of bijective mapsfrom G to (A,B, C), sayα : G → A, β : G → B, γ : G → Csuch that a · b = c if and only if α(a), β(b), γ(c) are three concurrent lines inPG(2,K), for any a, b, c ∈ G. If this is the case, the 3-net in PG(2,K) is said torealize the loop G.12 GÁBOR KORCHMÁROS AND GÁBOR P. NAGYFor the purpose of investigating 3-nets in PG(2,K), the groundfield K may beassumed to be algebraically closed. In order to present the key examples of embed-ded 3-nets, it is convenient to work with the dual concept. Formally, a dual 3-netof order n in PG(2,K) consists of a triple (Λ1,Λ2,Λ3) with Λ1,Λ2,Λ3 pairwisedisjoint point-sets of size n, called components, such that every line meeting twodistinct components meets each component in precisely one point. We notice thatfinite dual 3-nets are also called transversal designs.The following concepts and results have a detailed exposition in [10]. We saythat an embedded dual 3-net is algebraic, if its point set Λ1 ∪ Λ2 ∪Λ3 is containedin a cubic curve F . If F is reducible then we speak of pencil type, triangular type orconic-line type dual 3-net. Except for the pencil type, all algebraic (dual) 3-nets arecoordinatized by either a cyclic group or by a direct product of two cyclic groups.Finite dihedral groups can be realized by dual 3-nets of tetrahedron type; in thiscase the point set is contained in six lines joining four independent points. Finally,we mention that the quaternion group Q8 has an exceptional realization, cf. [16].In recent years, finite 3-nets realizing a group have been investigated also inconnection with complex line arrangements and resonance theory; see [2, 3, 5, 10,11, 12, 14, 17, 18]. The following almost complete classification of such 3-nets isproven in [10].Theorem 1.1. In the projective plane PG(2,K) defined over an algebraically closedfield K of characteristic p ≥ 0, let (Λ1,Λ2,Λ3) be a dual 3-net of order n ≥ 4 whichrealizes a group G. If either p = 0 or p > n then one of the following holds.(I) G is either cyclic or the direct product of two cyclic groups, and (Λ1,Λ2,Λ3)is algebraic.(II) G is dihedral and (Λ1,Λ2,Λ3) is of tetrahedron type.(III) G is the quaternion group of order 8.(IV) G has order 12 and is isomorphic to Alt4.(V) G has order 24 and is isomorphic to Sym4.(VI) G has order 60 and is isomorphic to Alt5.A computer aided exhaustive search shows that if p = 0 then (IV) (and hence(V), (VI)) does not occur; see [13]. It has been conjectured that this holds true inany characteristic.In this paper we focus on 3-nets in PG(2,K) which can be coordinatized by adiassociative loop G different from a group. Recall that a loop G is diassociative ifany subloop generated by two elements is a group. There are two important classesof diassociative loops: Moufang loops and Steiner loops. Moufang loops are loopssatisfying one (hence all) of the following identities.z(x(zy)) = ((zx)z)y, x(z(yz)) = ((xz)y)z, (zx)(yz) = (z(xy))z.In general, Moufang loops have a rich algebraic structure. This is not the case forSteiner loops. Steiner loops are diassociative loops of exponent two. Finite Steinerloops are in one-to-one connection with Steiner triple systems. For other classes ofdiassociative loops we refer to [9].Our results consist of three structural theorems on G.Theorem 1.2. In the projective plane PG(2,K) defined over an algebraically closedfield K of characteristic p ≥ 0, let (Λ1,Λ2,Λ3) be a dual 3-net of order n ≥ 4 whichrealizes a diassociative loop G different from a group. Let d be the maximum of the3-NETS REALIZING A DIASSOCIATIVE LOOP IN A PROJECTIVE PLANE 3orders of the elements in G, and suppose that d ≥ 4. If either p = 0 or p > n thenone of the following holds.(a) G has a unique subgroup H of order d. Moreover, each element not in His an involution, and two such involutions either commute or their productis in H.(b) d = 4, and G has a subgroup isomorphic to one of the groups Q8, Alt4.Theorem 1.3. With the same hypotheses as in Theorem 1.2, assume further thatG contains a subgroup isomorphic to Q8 but no subgroup isomorphic to Alt4. ThenG has a unique involution and the subgroup generated by any two non-commutingelements is isomorphic to Q8.It may be observed that a loop G as in Theorem 1.3 defines a Steiner triplesystem in a natural way, namely the points are subgroups of G of order 4 and theblocks are the subgroups isomorphic to Q8, the point-block incidence being the settheoretic inclusion.For a commutative loop G, neither (a) nor (b) of Theorem 1.2 can occur, andhence d ≤ 3. More precisely, the following result holds.Corollary 1.4. With the same hypotheses as in Theorem 1.2, assume further thatG is commutative. Then every non-trivial element in G has the same order, and Ghas exponent 2 or 3.The quaternion group Q8 has a counterpart in the class of Moufang loops. Let Obe the division ring of real octonions and let 1, e1, . . . , e7 be an orthonormal basis.The setO16 = {±1,±e1, . . . ,±e7}forms a Moufang loop with a unique involution −1 and 14 elements of order 4.(O16 is also called the Cayley loop of order 16.)Theorem 1.5. With the same hypotheses as in Theorem 1.2, assume further thatG is a Moufang loop. Then G contains either the octonion loop O16, or it has asubgroup isomorphic to Alt4.An interesting issue which appears to be rather difficult is the existence andconstruction of 3-nets in the classical projective plane PG(2,K) realizing a loopdifferent from a group. All such examples available in the literature are 3-nets oforder n = 5, 6, obtained by computer aided searches; see [16].2. Proof of Theorem 1.2Let Λ = (Λ1,Λ2,Λ3) be a 3-net of order n coordinatized by a diassociativeloop G but not by a group. Let g ∈ G be an element whose order is d, and letΨ = (Ψ1,Ψ2,Ψ3) be the 3-net (subnet of Λ) coordinatized by 〈g〉. Take any elementh ∈ G not lying in the cyclic group generated by g, and consider the subgroup Hgenerated by g and h. Obviously, H is not a cyclic group. Since H is a subloopof G, H also realizes a 3-net ∆ = (∆1,∆2,∆3) in PG(2,K). The classification[10, Theorem 1.1] applies to H and yields one of the cases below, apart from thesporadic cases. Therefore, dismissing (b), we have that either(i) H is the direct product of two cyclic subgroups, and ∆1 ∪∆2 ∪∆3 lies ona plane non-singular cubic curve F3, or(ii) H is a dihedral group, and ∆ is of tetrahedron type.4 GÁBOR KORCHMÁROS AND GÁBOR P. NAGYWe first investigate case (i). As G is not a group, it must contain an element u 6∈ H .Replacing h,H by u, U = 〈g, u〉 in the above argument shows that U realizes a 3-netΦ = (Φ1 ∪ Φ2 ∪ Φ3) in PG(2,K), and that U is either the direct product of twocyclic groups, or it is a dihedral group. The latter case here cannot actually occur.To show this, observe that if U is dihedral then the maximality of d implies that uis an (involutory) element lying in some coset of 〈g〉. Since Φ is of tetrahedron typein this case, we have that Ψ is a triangular 3-net in PG(2,K) of order d. But thisis impossible in our case, since the points of Ψ lie on F3. In fact, F3 is non-singularwhile Ψ1 consists of d > 3 collinear points. Therefore, U is a direct product of twocyclic groups and Φ1 ∪ Φ2 ∪ Φ3 lies on a non-singular plane cubic curve F1. Theintersection F3 ∩ F1 contains all points of Ψ. Since d > 3, this yields F3 = F1.Since u denotes any element of G outside H , it turns out that Λ1 ∪Λ2 ∪Λ3 lies onF3. But then G itself is a group, the direct product of two cyclic groups. Therefore,(i) cannot actually occur.In case (ii), we may assume that H1 = 〈g, h1〉 is dihedral for any h1 ∈ G \ 〈g〉.Therefore, h1 is an involution. Moreover, if h2 is another involution in G, theneither h2 commutes with h1, or h2 lies H1.3. Proof of Theorem 1.3From the definition of a dual 3-net, there is a triple of bijective maps from Gto (Λ1,Λ2,Λ3), say α, β, γ respectively such that a · b = c in G if and only ifα(a), β(b), γ(c) are three collinear points in PG(2,K), for any a, b, c ∈ G.Choose two elements g1, g2 ∈ G which generate a subgroup H isomorphic to Q8.We remark that case (a) in Theorem 1.2 cannot occur since both g and h haveorder 4. Therefore, d = 4. Set g3 = g1g2; then 〈g1〉, 〈g2〉, 〈g3〉 are the three cyclicsubgroups of order 4 in H . Take an element u ∈ G not lying in H .Assume that u is an involution. For i = 1, 2, 3, the group Ui = 〈u, gi〉 contains atleast two distinct involutions, and hence it is not isomorphic to Q8. From Theorem1.1 applied to Ui, we deduce that either Ui is dihedral, or abelian.We first investigate the case when all Ui’s are abelian. Clearly, Ui = 〈gi〉× 〈u〉 ∼=C4×C2, and case (a) of Theorem 1.2 holds for the sub 3-net (∆i1,∆i2,∆i3) realizingUi. Let Fi be the cubic curve containing ∆i1 ∪∆i2 ∪∆i3. Since Ui is not cyclic, Fiis nonsingular. These three sub 3-nets of order 8 share a sub 3-net of order 4, say(Ω1,Ω2,Ω3), realizing the group T = 〈u, g21 = g22〉. Since |F1 ∩ F2 ∩ F3| ≥ 12 > 9,this yields that F1 = F2 = F3. But then the sub 3-net realizing H lies on F1 acontradiction since Q8 is not abelian.Assume now that U1, U2 are abelian and U3 dihedral. With the same argument,the dual sub 3-nets realizing U1, U2 are contained in the nonsingular cubic curve F .The dual sub 3-net realizing U3 is of tetrahedron type, which means that the fourpoints of α(〈g3〉) are collinear. The triple (α(〈g3〉), β(H \〈g3〉), γ(H \〈g3〉)) is a dual3-net realizing 〈g3〉. On the one hand, the 8 points of β(H \ 〈g3〉) ∪ γ(H \ 〈g3〉) arecontained in a (possibly degenerate) conic C, see [2, Theorem 5.1]. On the otherhand, H \ 〈g3〉 ⊂ 〈g1〉 ∪ 〈g2〉 ⊂ U1 ∪U2. This implies β(H \ 〈g3〉)∪ γ(H \ 〈g3〉) ⊂ Fand |F ∩ C| ≥ 8, a contradiction.Assume that U1, U2 are dihedral. Hence the dual sub 3-nets realizing U1, U2 areof tetrahedron type yielding that the four points of α(〈g1〉) and the four points ofα(〈g2〉) are contained in the lines ℓ1, ℓ2, respectively. However, α(1), α(g21 = g22) ∈ℓ1∩ ℓ2, thus, ℓ1 = ℓ2. Similarly, the six points of β(〈g1〉∪ 〈g2〉) and the six points of3-NETS REALIZING A DIASSOCIATIVE LOOP IN A PROJECTIVE PLANE 5γ(〈g1〉 ∪ 〈g2〉) are contained in the lines m,m′, respectively. If U3 is dihedral, thenthe dual sub 3-net realizing H is contained in ℓ1∪m∪m′, which is impossible sinceH is not cyclic. If U3 is abelian, then the sub 3-net realizing it is contained in thenonsingular cubic curve F . The second component of its sub 3-net (α(〈g3〉), β(H \〈g3〉), γ(H \ 〈g3〉)) is contained in m, hence |m ∩ F| ≥ 4, a contradiction.Assume that u has order 3. Then U is neither a dihedral group nor isomorphicto Q8. From Theorem 1.1, U = 〈g〉 × 〈u〉 and hence U is a cyclic group of order 12contradicting the remark at the beginning about case (a) in Theorem 1.2.Therefore, G contains just one involution v, and if u 6= v then u has order 4. Letu1, u2 ∈ G be any two distinct elements other than v. Since U contains no elementof order 3, U = 〈u1, u2〉 is a 2-group of exponent 4 containing a unique involution.Since U has order bigger than 4, the only possibility is U ∼= Q8.Remark 3.1. Let S be the Steiner loop of order 10 corresponding to the Steinertriple system AG(2, 3). S has a central extension Q of order 20 all proper subloopsare isomorphic to C2, C4, C2 × C4, or Q8. In particular, Q is diassociative. ByTheorem 1.3, Q has no projective realization despite all its subloops have.4. Proof of Theorem 1.5We start with three important facts on Moufang loops of small exponent. First,as diassociative loops of exponent 2 are commutative, Moufang loops of exponent2 are elementary abelian groups. Second, by [6, Corollary 1] finite Moufang loopsof exponent 3 are nilpotent. This implies that any proper finite Moufang loop ofexponent 3 contains a subloop of order 27. The classification of small Moufangloops [8] shows that Moufang loops of order 27 are groups. Thus, if G is a Moufangloop of exponent 3 then it contains a subgroup H of order 27. Since no such groupH has a realization in PG(2,K) by Theorem 1.1, we have a contradiction.Let us assume that G has an element g of order d > 4. Put U = 〈g〉. ByTheorem 1.2, any h ∈ G \U has order 2 and 〈U, h〉 is a dihedral group of order 2d.In particular, on the one hand, hU = Uh, and on the other hand, the involutionsgenerate G. [7, Theorem 1] implies that U is a normal subloop of G. For anysubset X of G, let Λi(X) denote the points of Λi, indexed by the elements of X .[10, Proposition 22] implies that any of the sets Λi(U), Λi(Uh) is contained in aline.Choose an element h ∈ G \ U . Then we have four points P,Q,R, S such thatthe point sets Λ1(U),Λ2(U),Λ3(U),Λ1(Uh),Λ2(Uh),Λ3(Uh) are contained in thelines QR,RS, PR, SP, SQ, PQ. In fact, the points P,Q,R, S are the vertices of thetetrahedron type dual 3-net, corresponding to the dihedral group 〈U, h〉. Only thevertex S depends on the choice of h; S = Sh.Choose elements h1, h2 ∈ G\U such that 〈U, h1, h2〉 is a non-associative subloopof G. Let P,Q,R, Sh1 , Sh2 , Sh1h2 be the vertices of the tetrahedron type dual netsof 〈U, h1〉, 〈U, h2〉 and 〈U, h1h2〉. The setsΛ1(Uh1),Λ2(Uh2),Λ3(Uh1h2)of points form a triangular dual 3-net. [10, Proposition 10] implies Sh1 = Sh2 =Sh1h2 , a contradiction.Finally, assume that G has no subgroup isomorphic to Alt4. By Theorem 1.3,G has a unique (central) involution u. As two non-commuting elements generatea subloop isomorphic to Q8, the factor G/〈u〉 is an elementary abelian 2-group.6 GÁBOR KORCHMÁROS AND GÁBOR P. NAGYThus, G contains a non-associative subloop S of order 16 with a unique involution.Using the classification of small Moufang loops in [8], S ∼= O16 follows.5. acknowledgementThe work has been carried out within the Project PRIN (MIUR, Italy) andGNSAGA. The publication is supported by the European Union and co-funded bythe European Social Fund. Project title: Telemedicine-focused research activitieson the field of Mathematics, Informatics and Medical sciences. Project number:TAMOP-4.2.2.A-11/1/KONV-2012-0073.References[1] A. Barlotti and K. Strambach, The geometry of binary systems, Adv. Math. 72 (1988), 1-58.[2] A. Blokhuis, G. Korchmáros and F. Mazzocca, On the structure of 3-net embedded in aprojective plane, J. Combin. Theory Ser.A 118 (2011) 1228-1238.[3] N. Bogya, G. Korchmáros and G.P. Nagy, Classification of k-nets, European J. Combin. 48(2015), 177–185.[4] R.H. Bruck, A survey of binary systems, Ergebnisse der Mathematik und ihrer Grenzgebiete.Neue Folge, Heft 20. Reihe: Gruppentheorie Springer Verlag, Berlin-Gttingen-Heidelberg1958 viii+185 pp.[5] M. Falk and S. Yuzvinsky, Multinets, resonance varieties, and pencils of plane curves, Compos.Math. 143 (2007), 1069–1088.[6] G. Glauberman and C.R.B. Wright, Nilpotence of finite Moufang 2-loops, J. Algebra 8 (1968),415-417.[7] S. M. Gagola III, When are inner mapping groups generated by conjugation maps? Arch.Math. (Basel) 101 (2013), 207–212.[8] Goodaire, E. G., May, S., Raman, M., 1999. The Moufang Loops of Order less than 64, NovaScience Publishers.[9] M. K. Kinyon, K. Kunen and J. D. Phillips, A generalization of Moufang and Steiner loops,Algebra Universalis 48 (2002), 81–101.[10] G. Korchmáros, G.P. Nagy and N. Pace, 3-nets realizing a group in a projective plane, J.Algebraic Comb. 39 (2014), 939–966.[11] G. Korchmáros, G.P. Nagy and N. Pace, k-nets embedded in a projective plane over a field,to appear in Combinatorica, 35 (2015), 63-74.[12] Á. Miguel and M. Buzunáriz, A description of the resonance variety of a line combinatoricsvia combinatorial pencils, Graphs and Combinatorics 25 (2009), 469-488.[13] G. Nagy and N. Pace, Some computational results on small 3-nets embedded in a projectiveplane over a field, J. Comb. Theory, Ser. A 120 (2013), 1632–1641.[14] J.V. Pereira and S. Yuzvinsky, Completely reducible hypersurfaces in a pencil, Adv. Math.219 (2008), 672-688.[15] J. Stipins, Old and new examples of k-nets in P2, arXiv:1104.4439v1.[16] G. Urzúa, On line arrangements with applications to 3-nets, Adv. Geom. 10 (2010), 287-310.[17] S. Yuzvinsky, Realization of finite abelian groups by nets in P2, Compos. Math. 140 (2004),1614–1624.[18] S. Yuzvinsky, A new bound on the number of special fibers in a pencil of curves, Proc. Amer.Math. Soc. 137 (2009), 1641–1648.Dipartimento di Matematica, Informatica ed Economia, Università della Basilicata,Contrada Macchia Romana, 85100 Potenza, ItalyE-mail address: gabor.korchmaros@unibas.itBolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hun-garyMTA-ELTE Geometric and Algebraic Combinatorics Research Group, Pázmány P.sétány 1/c, H-1117 Budapest, HungaryE-mail address: nagyg@math.u-szeged.hu

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Nagy, Gábor Péter

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3-nets realizing a diassociative loop in a projective plane

3-nets realizing a diassociative loop in a projective plane

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