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LINEAR GROUPS AS RIGHT MULTIPLICATION GROUPS of QUASIFIELDS 3 Group T (Π) of Translations

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LINEAR GROUPS AS RIGHT MULTIPLICATION GROUPS of QUASIFIELDS 3 Group T (Π) of Translations LINEAR GROUPS AS RIGHT MULTIPLICATION GROUPS OF QUASIFIELDS GABOR´ P. NAGY Abstract. For quasifields, the concept of parastrophy is slightly weaker than isotopy. Parastrophic quasifields yield isomorphic translation planes but not conversely. We investigate the right multiplication groups of fi- nite quasifields. We classify all quasifields having an exceptional finite transitive linear group as right multiplication group. The classification is up to parastrophy, which turns out to be the same as up to the iso- morphism of the corresponding translation planes. 1. Introduction A translation plane is often represented by an algebraic structure called a quasifield. Many properties of the translation plane can be most easily understood by looking at the appropriate quasifield. However, isomorphic translation planes can be represented by nonisomorphic quasifields. Further- more, the collineations do not always have a nice representation in terms of operations in the quasifields. Let p be a prime number and (Q, +, ·) be a quasifield of finite order pn. Fn We identify (Q, +) with the vector group ( p , +). With respect to the multiplication, the set Q∗ of nonzero elements of Q form a loop. The right multiplication maps of Q are the maps Ra : Q → Q, xRa = x · a, where Fn a, x ∈ Q. By the right distributive law, Ra is a linear map of Q = p . Clearly, R0 is the zero map. If a 6= 0 then Ra ∈ GL(n,p). In geometric context, the set of right translations are also called the slope set or the spread set of the quasifield Q, cf. [6, Chapter 5]. The right multiplication group RMlt(Q) of the quasifield Q is the linear group generated by the nonzero right multiplication maps. It is immediate to see that RMlt(Q) is a transitive linear group, that is, it acts transitively Fn arXiv:1210.1652v1 [math.CO] 5 Oct 2012 on the set of nonzero vectors of Q = p . The complete classification of finite transitive linear groups is known, the proof relies on the classification theorem of finite simple groups. Roughly speaking, there are four infinite classes and 27 exceptional constructions of finite transitive linear groups. The first question this paper deals with asks which finite transitive linear group can occur as right multiplication group of a finite quasifield. It turns out that some of the exceptional finite transitive linear groups may happen to be the right multiplication group of a quasifield. Since these groups are 2010 Mathematics Subject Classification. 51A40, 05B25, 20N05. Key words and phrases. Quasifield, Right multiplication group, Translation plane, Au- totopism, Loop. The author is a Janos Bolyai Research Fellow. Supported by TAMOP-4.2.2/B-10/1- 2010-0012 project of Hungary. 1 2 GABOR´ P. NAGY relatively small, in the second part of the paper, we are able to give an explicit classification of all quasifields whose right multiplication group is an exceptional finite linear group. These results are obtained with computer calculations using the computer algebra system GAP4 [2] and the program CLIQUER [13]. Right multiplication groups of quasifields have not been studied inten- sively. The most important paper in this field is [8] by M. Kallaher, contain- ing results about finite quasifields with solvable right multiplication groups. Another related paper is [11] which deals with the group generated by the left and right multiplication maps of finite semifields. Finally, we notice that our results can be interpreted in the language of the theory of finite loops, as well. Loops arise naturally in geometry when coordinatizing point-line incidence structures. Most importantly, the multiplicative structure (Q∗, ·) of Q is a loop. In fact, any finite loop Qˆ gives rise to a quasifield, provided the right multiplication maps of Qˆ generate a group which is permutation isomorphic to a subgroup of GL(n,q), where the latter is considered as a permutation group on the nonzero vectors of Fn q . Therefore in this paper, we investigate loops whose right multiplication groups are contained is some finite linear group GL(n,q). There are many excellent surveys and monographs on translation planes and quasifields, see [4, 6, 9] and the references therein. Our computational methods have similarities with those in [1, 3]. 2. Translation planes, spreads and quasifields For the shake of completeness, in this section we repeat the definitions of concepts which are standard in the theory of translation planes. More- over, we briefly explain the relations between the automorphisms of these mathematical objects. Let Π be a finite projective plane. The line ℓ of Π is a translation line if the translation group with respect to ℓ acts transitively (hence regularly) on the set of points Π \ ℓ. Assume ℓ to be a translation line and let Πℓ be the affine plane obtained from Π with ℓ as the line at infinity. Then Πℓ is a translation plane. By the theorems of Skornyakov-San Soucie and Artin-Zorn [4, Theorem 6.18 and 6.20], Π is either Desarguesian or contains at most one translation line. This means that two finite translation planes are isomorphic if and only if the corresponding projective planes are. The relation between translation planes and quasifields is usually ex- plained using the notion of (vector space) spreads. Definition 2.1. Let V be a vector space over the field F . We say that the collection σ of subspaces is a spread if (1) A, B ∈ σ, A 6= B then V = A ⊕ B, and (2) every nonzero vector x ∈ V lies in a unique member of σ. The members of σ are the components of the spread. If σ is a spread in V then Andr´e’s construction yields a translation plane Π(σ) by setting V as the set of points and the translates of the components of σ as the set of lines of the affine plane. Conversely, if Π is a finite translation plane with origin O then we identify the point set of Π with the LINEAR GROUPS AS RIGHT MULTIPLICATION GROUPS OF QUASIFIELDS 3 group T (Π) of translations. As T (Π) is an elementary Abelian p-group, Π becomes a vector space over (some extension of) Fp and the lines through O are subspaces forming a spread σ(Π). Andr´e’s construction implies a natural identification of the components of the spread with the parallel classes of the affine lines, and the points at infinity of the corresponding affine plane. The approach by spreads has many advantages. For us, the most impor- tant one is that they allow explicit computations in the group of collineation of the translation plane. Let Π be a nondesarguesian translation plane and let us denote by T (Π) the group of translations of Π. The full group Aut(Π) of collineations contains T (Π) as a normal subgroup. Up to isomorphy, we can choose a unique point of origin O in Π. The stabilizer CO(Π) of O in Aut(Π) is the translation complement of Π with respect to O. The full group Aut(Π) of collineations is the semidirect product of T (Π) and CO(Π). In particular, CO(Π) has the structure of a linear group of V . By [6, Theo- rem 2.27], the collineation group CO(Π) is essentially the same as the group of automorphisms of the associated spread, where the latter is defined as follows. Definition 2.2. Let σ be a spread in the vector space V . The automor- phism group Aut(σ) consists of the additive mappings of V that permutes the components of σ among themselves. As the translations act trivially on the infinite line, the permutation action of Aut(σ) is equivalent with the action of Aut(Π) on the line at infinity. Spreads are usually represented by a spread set of matrices. Fix the components A, B of the spread σ with underlying vector space V . The direct sum decomposition V = A ⊕ B defines the projections pA : V → A, pB : V → B. As for any component C ∈ σ \ {A, B} we have A ∩ C = B ∩C = 0, the restrictions of pA and pB to C are bijections C → A, C → B. −1 Therefore, the map uC : A → B, xuC = (xpA )pB is a linear bijection from Fk A to B. When identifying A, B with p, uC can be given in matrix form UC . The set S(σ) = {UC | C ∈ σ} is called the spread set of matrices representing σ relative to axes (A, B). This representation depends on the choice of A, B ∈ σ. It is also possible to think at a spread set as the collection ′ −1 S (σ)= {uDuC | D ∈ σ} of linear maps A → A, with fixed C. A spread set S ⊆ hom(A, B) can be characterized by the following prop- erty: For any elements x ∈ A\{0}, y ∈ B \{0}, there is a unique map u ∈ S such that xu = y. Indeed, if S = S(σ) then u = uC where C is the unique component containing x ⊕ y. Conversely, S defines the spread σ(S)= {0 ⊕ B, A ⊕ 0} ∪ {{x ⊕ xu | x ∈ A} | u ∈ S} with underlying vector space V = A ⊕ B. Definition 2.3. The autotopism group of the spread set S of k × k ma- trices over F consists of the pairs (T,U) ∈ GL(k, F ) × GL(k, F ) such that T −1SU = S. [6, Theorems 5.10] says that autotopisms of spread sets relative to axes (A, B) and automorphism of spreads fixing the components A, B are essen- tially the same. 4 GABOR´ P. NAGY By fixing a nondegenerate quadrangle o, e, x, y, any projective plane can be coordinatized by a planar ternary ring (PTR), see [4].
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