Des. Codes Cryptogr. DOI 10.1007/s10623-015-0061-y
Transitive hyperovals Benjamin C. Cooper1 · Tim Penttila1
Received: 1 October 2014 / Revised: 28 February 2015 / Accepted: 2 March 2015 © Springer Science+Business Media New York 2015
Abstract We complete the classification of transitive hyperovals with groups of order divisible by four. Keywords Finite partial geometries (general) · Nets · Partial spreads · Blocking sets · Ovals · k-arcs Mathematics Subject Classification
51E14 · 51E21
1 Introduction In 1987, Biliotti and Korchmaros [2] showed that a transitive hyperoval of a projective plane of even order that admits a group of order divisible by four is either a regular hyperoval in a Desarguesian plane of order two or four or is in a plane of order 16 and has group of order at most 144. In 2005, Sonnino [7] showed that a transitive hyperoval of a projective plane of order 16 with a group of order 144 is necessarily the hyperoval of the Desarguesian plane of order 16 constructed by Lunelli and Sce [6] in 1958. Here we rule out the remaining cases, completing the proof of our main theorem. Theorem 1 A transitive hyperoval of a projective plane of even order that admits a group of order divisible by four is either a regular hyperoval in a Desarguesian plane of order two or four or the Lunelli–Sce hyperoval of the Desarguesian plane of order 16.
This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Finite Geometries”.
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Benjamin C. Cooper
[email protected] Tim Penttila
[email protected]
1
Colorado State University, Fort Collins, CO, USA
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2 Background A projective plane π is an incidence structure of points and lines such that every pair of points is incident with a unique line, every pair of lines is incident with a unique point and there exist four points, no three collinear. A projective plane is finite if it has only finitely many points, in which case, it has an order: there is an integer n ≥ 2 such that every line is incident with n + 1 points. In a projective plane of order n, there are n 2 + n + 1 points, n 2 + n + 1 lines, and every point is incident with n + 1 lines. An example of a finite projective plane is PG(2, q) which has points the one-dimensional subspaces of G F(q)3 , lines the 2-dimensional subspaces of G F(q)3 and incidence inclusion. PG(2, q) has order q. These projective planes are characterized amongst finite projective planes by Desargues’ theorem, and are therefore called Desarguesian. A subplane of a projective plane π is a subset π0 of the point set of π such that, together with the lines that are incident with more than one point of π0 , and with the restricted incidence, π0 is itself a projective plane. The lines incident with more than one point of π0 are called the lines of π0 . π0 is proper if π0 = π. Theorem 2 (Baer [1]) Let π0 be a proper subplane of order m of a projective plane of order n. Then m 2 ≤ n and equality implies that every point of π that is not a point of π0 is incident with a unique line of π0 . A subplane of order m of a plane of order m 2 is called a Baer subplane. PG(2, q) is a Baer subplane of PG(2, q 2 ). A k-arc of a projective plane is a set of k points, no three collinear. Theorem 3 (Bose [3]) A k-arc in a projective plane of order n has k ≤ n + 2. Equality implies n is even. An (n + 2)-arc in a projective plane of order n is called a hyperoval. Lines meeting a hyperoval are called secant; otherwise external. The set {(1, t, t 2 ) : t ∈ G F(q)} ∪ {(0, 1, 0), (0, 0, 1)} is a hyperoval of PG(2, q), called the regular hyperoval. Another hyperoval plays a leading role in this paper; it was discovered by Lunelli and Sce via a computer search in 1958. Theorem 4 [6] {(1, t, t 12 +t 10 +η11 t 8 +t 6 +η2 t 4 +η9 t 2 ) : t ∈ G F(q)}∪{(0, 1, 0), (0, 0, 1)}, where η ∈ G F(16) satisfies η4 = η + 1, is a hyperoval of PG(2, 16) called the Lunelli–Sce hyperoval. A collineation of a projective plane π is a permutation of the points and lines taking points to points and lines to lines and preserving incidence. The collineations of π from a group Aut (π) under composition. Theorem 5 (Fundamental theorem of projective geometry) Aut (PG(2, q)) = PΓ L(3, q). Our agenda is the study of hyperovals of projective planes π whose stabiliser G in Aut (π) acts transitively on the hyperoval. If π has order n, this implies that n + 2 divides G, so that |G| is even. We will need a further hypothesis in order to gain traction: we will suppose that |G| is divisible by four. The difficulties in classifying such hyperovals all concern nonDesarguesian planes:
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Theorem 6 (Korchmaros [5]) The transitive hyperovals of finite Desarguesian planes are precisely the regular hyperovals in PG(2, 2) and PG(2, 4) and the Lunelli–Sce hyperoval in PG(2, 16). Since the plane in which the hyperoval lives is unknown, it is convenient to shift attention to an associated structure. An abstract hyperoval of order n on X can be defined to be a set A(X ) of fixed-point-free involutory permutations of a set X of cardinality n+2 ≥ 4 such that whenever a1 , a2 , b1 , b2 ∈ X are distinct, there exists a unique σ ∈ A(X ) with σ (a1 ) = a2 and σ (b1 ) = b2 . Each hyperoval X of a projective plane of order n gives an abstract hyperoval with the involutions being in one-to-one correspondence with the points of the plane not on the hyperoval; the involution corresponding to a point P maps Q ∈ X to R, where P Q ∩ X = {Q, R}. We will need the classification of transitive permutation groups of degree 18. Theorem 7 (Hulpke [4]) There are 983 transitive groups of degree 18.
3 Prior results Theorem 8 (Biliotti–Korchmaros [2]) A transitive hyperoval in a finite projective plane of order n is either a regular hyperovals in PG(2, 2) or PG(2, 4) or n = 16. If n = 16, then the order of the group G of the hyperoval divides 144, the group fixes a Baer subplane π0 disjoint from the hyperoval (and the pointwise stabilizer G (π0 ) of π0 in G has order at most two) and G centralizes a unitary polarity of π0 and acts transitively on the absolute points of that polarity in π0 . Moreover, G contains exactly nine involutions fixing two points of the hyperoval. Theorem 9 (Sonnino [7]) A transitive hyperoval in a finite projective plane of order 16 with a group of order 144 is a Lunelli–Sce hyperoval in PG(2, 16). We will follow Sonnino’s methods closely, so it is necessary to describe them. The description in Theorem 8 of the group G of the hyperoval means that either G is the unique up to conjugacy subgroup of PΓ U (3, 4) of order 144 or the kernel of the action on the Baer subplane π0 has order two and G π0 is AG L(1, 9) or the semidirect product of C3 × C3 by Q 8 . Consideration of the possible transitive actions of degree 18 of these groups, it is feasible to mount a computer search for abstract hyperovals of order 18 admitting these actions. Each such transitive action on a set X induces an action on the fixed-point-free involutions of degree 18, and any such abstract hyperoval A(X ) must be a union of orbits of this induced action. Many orbits O fail to satisfy the condition (necessary to be a subset of an abstract hyperoval) that whenever a1 , a2 , b1 , b2 ∈ X are distinct, there is at most one σ ∈ O with σ (a1 ) = a2 and σ (b1 ) = b2 . Define a graph with vertices the orbits satisfying this condition and edges the unions O of a pair of vertices satisfying this condition. Then A(X ) corresponds to a clique of this graph; and mounting a clique search leads to all abstract hyperovals arising from a transitive hyperoval in a finite projective plane of order 16 with a group of order 144. It turns out that they all arise from a Lunelli–Sce hyperoval in PG(2, 16) and that such an abstract hyperoval embeds in a unique projective plane of order 16.
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4 Our methods Sonnino’s methods need to be sharpened a little to make the remaining cases computationally feasible. It turns out that good use can be made of the invariant Baer subplane. Lemma 1 Let A(X ) be an abstract hyperoval of order 16 arising from a transitive hyperoval X in a finite projective plane π of order 16 with group G. Then there is an orbit U of G on A(X ) of size nine and a union U of orbits of G on A(X ) of size 12 such that, if B = U ∩ U then there is a set L of nine subsets of B of size five with |Fi x(σ τ )| = 2 for σ = τ in and a set M of 12 subsets m of B of size five with |Fi x(σ τ )| = 0 for σ = τ in m with the further property that the incidence structure with point set B, line set L ∪ M and incidence set membership is a projective plane π0 of order four and U is a unital of π0 . Proof By Theorem 8, G leaves invariant a Baer subplane π0 of π which is disjoint from X . The points of π not on X correspond to elements of A(X ) and the permutations groups (G, π \ X ) and (G, A(X )) are permutationally isomorphic via this correspondence. Moreover, π0 is a union of G-orbits corresponding to the nine absolute points of the G-invariant unitary polarity (which form a unital of π0 ) and of G-orbits corresponding to the 12 non-absolute points of the G-invariant unitary polarity. Every point of the hyperoval X lies on a unique line of π0 ; this line is secant to X . Moreover, the points on this line and not on X correspond to element of A(X ) interchanging the points on the line on X ; and hence their pairwise products fix these points. These nine lines of π0 secant to X give the set L via the correspondence between points not on X and elements of A(X ). The remaining 12 lines of π0 give the set M via the correspondence between points not on X and elements of A(X ); since they are external to X , the pairwise products of distinct elements corresponding to such a line are fixed-point-free.
Like Sonnino, our programs were implemented in the computer algebra system Magma. It turns out to be far more efficient to search for the possible subsets U and U arising from Lemma 1 of an abstract hyperoval of order 16 arising from a transitive hyperoval X in a finite projective plane π of order 16 first, and then to seek other orbits of the underlying group compatible with U ∪ U . We ran through the list of transitive subgroups of degree 18 acting on X = {1, . . . , 18}, first checking the conditions given by Theorem 8 for the groups, and then, for each of the surviving groups G, finding the orbits O on fixed-point-free involutions such that whenever a1 , a2 , b1 , b2 ∈ X are distinct, there is at most one σ ∈ O with σ (a1 ) = a2 and σ (b1 ) = b2 . The next step was to find the projective planes π0 of order four given by Lemma 1 invariant under G. Now, for each π0 , we find the set V of orbits O on fixed-point-free involutions such that whenever a1 , a2 , b1 , b2 ∈ X are distinct, there is at most one σ ∈ O ∪ π0 with σ (a1 ) = a2 and σ (b1 ) = b2 . Now we find the set E of pairs {O, O } with O, O ∈ V such that there is at most one σ ∈ O ∪ O ∪ π0 with σ (a1 ) = a2 and σ (b1 ) = b2 . Now the cliques of the graph Γ = (V, E) are found, where the union of the elements of the clique contains 255 fixed-point-free involutions (255 is the size of an abstract hyperoval of order 16). It turned out that all the abstract hyperovals arising from the search had a group of order 144, and so, by Theorem 9, X is a Lunelli–Sce hyperoval and π is Desarguesian. Together with Theorem 8, this proves Theorem 1. In more detail, 39 of the 983 transitive groups of degree 18 have orders divisible by 36 and dividing 144. By Theorem 9, we could restrict our attention to the 24 groups of order 36 or 72. Restricting our attention to groups contain no proper transitive subgroup of order divisible by four reduces this list to nine groups. Applying Theorem 8, can reduce this still further to five groups, as the remaining groups are neither isomorphic to a subgroup of PΓ U (3, 4)
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nor have a normal subgroup of order two such that the quotient group is isomorphic to a subgroup of PΓ U (3, 4). One of the remaining groups does not contain nine involutions with two fixed points, so can be eliminated by Theorem 8. The four surviving groups were TransitiveGroup(18,i), for i = 9, 10, 12, 28, in the implementation of Hulpke’s result on Magma. Each of these was fed into our algorithm. By using the induced action on fixed-point-free involutions, it was possible to calculate the stabilizer of each abstract hyperoval that arose from running our software, and to check that its preimage in Sym(X ) was permutationally isomorphic to the group of the Lunelli–Sce hyperoval acting on the points of the Lunelli–Sce hyperoval. Now Theorem 9 applies; the hyperoval X is a Lunelli–Sce hyperoval and π is Desarguesian.
References 1. Baer R: Projectivities with fixed points on every line of the plane. Bull. Am. Math. Soc. 52, 273–286 (1946). 2. Biliotti M., Korchmaros G.: Hyperovals with a transitive collineation group. Geom. Dedicata 24, 269–281 (1987). 3. Bose R.C.: Mathematical theory of the symmetrical factorial design. Sankhy¯a 8, 107–166 (1947). 4. Hulpke A.: Constructing transitive permutation groups. J. Symb. Comput. 39, 1–30 (2005). 5. Korchmaros G.: Collineation groups transitive on the points of an oval [(q + 2)-arc] of S2,q for q even. Atti Sem. Mat. Fis. Univ. Modena 27, 89–105 (1978). 6. Lunelli L., Sce M.: q-Archi completi dei piani desarguesiani di rango 8 e 16. Centro Calc. Num. Politec. Milano (1958). 7. Sonnino A.: Transitive hyperovals in finite projective planes. Australas. J. Comb. 33, 335–347 (2005).
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