Des. Codes Cryptogr. (2008) 47:237–242 DOI 10.1007/s10623-007-9146-6

Generalized quadrangles admitting a sharply transitive Heisenberg group S. De Winter · K. Thas

Received: 26 October 2006 / Revised: 16 October 2007 / Accepted: 17 October 2007 / Published online: 29 November 2007 © Springer Science+Business Media, LLC 2007

Abstract All known finite generalized quadrangles that admit an automorphism group acting sharply transitively on their point set arise by Payne derivation from thick elation generalized quadrangles of order s with a regular point. In these examples only two groups occur: elementary abelian groups of even order and odd order Heisenberg groups of dimension 3. In [2] the authors determined all generalized quadrangles admitting an abelian group with a sharply transitive point action. Here, we classify thick finite generalized quadrangles admitting an odd order Heisenberg group of dimension 3 acting sharply transitively on the points. In fact our more general result comes close to a complete solution of classifying odd order Singer p-groups. Keywords

Generalized quadrangle · Singer group · Heisenberg group · Payne derivation

AMS Classifications

05B10 · 05B25 · 05E20 · 20B25 · 51E12 · 51E20

1 Singer generalized quadrangles A Singer group (w.r.t. points) of a point-line incidence geometry is an automorphism group of the geometry acting sharply transitively on its points. Before proceeding, recall that a finite (thick)1 generalized n-gon S of order (s, t), n ≥ 2, s > 1, t > 1 (where all the parameters are finite), is a 1 − (v, s + 1, t + 1) design whose incidence graph has girth 2n and diameter n. Whenever s = t the generalized n-gon is said to have order s. The (finite) 1 Only thick generalized n-gons are considered in this note, even when we do not mention this explicitly in

statements, etc. S. De Winter (B) · K. Thas Department of Pure Mathematics and Computer Algebra, Ghent University, Krijgslaan 281, S22, 9000, Ghent, Belgium e-mail: [email protected] K. Thas e-mail: [email protected]

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generalized 3-gons are precisely the (finite) projective planes. The theory of Singer groups for projective planes has been especially popular and productive. Many questions concerning Singer groups stand among the truly fundamental ones in the theory of projective planes, the most notable being perhaps the classification of planes admitting an abelian Singer group. Conjecturally, those should always be Desarguesian. In the past 15 years, the question has been posed several times whether there are fruitful Singer group theories for other types of (building like) geometries, especially for the other generalized n-gons. For generalized 4-gons, or also “generalized quadrangles” (GQs), such a theory was initiated by D. Ghinelli in [3], where it was shown that a finite GQ of order s cannot admit an abelian Singer group. In [2] the authors further developed the theory by determining all GQs admitting an abelian Singer group. In fact, they show that a GQ admitting an abelian Singer group G must always arise by “Payne derivation” from a translation GQ of even order s. It follows that G is necessarily elementary abelian. This paper can be seen as part of a larger project in which the authors are currently involved, the eventual aim of which is the complete determination of all GQs admitting a Singer group. This article is a continuation and completion of [2], in the sense that, as a first step in the project, we determine all GQs that admit a Singer group G for those G that are known to act as Singer groups on some GQ. The only known such groups are the (elementary) abelian groups (of even order), dealt with in [2], and the odd order Heisenberg groups of dimension 3 over GF(q), which are the subject of this paper. Throughout the paper standard notation on GQs will be used (see the monograph [8]). We mention the
concepts and theorems. In a GQ Q of order s a point p is said to be
following regular if
{p, r}⊥⊥
= s + 1 for any point r  = p; it is said to be a center of symmetry if there exists a group G of size s of automorphisms of Q fixing all points of p ⊥ and acting semiregularly on the points not collinear with p. It is easily checked that each center of symmetry is a regular point. The following beautiful construction of GQs is due to Payne [5]. Let Q be a GQ of order s with a regular point p. Define the point-line incidence geometry P = P (Q, p) as follows. • The points of P are the points of Q not collinear with p. • The lines of P are the lines of Q not through p together with all sets {p, r}⊥⊥ \ {p}, where p  ∼ r. We have the following theorem. Theorem 1.1 (S. E. Payne [5]) The point-line geometry P is a GQ of order (s − 1, s + 1). The GQ P is the so-called Payne derived GQ of Q (with respect to p). A spread T of a GQ Q of order (s, T of the point set of Q into lines. A

t) is a partition spread T is called regular1 provided
{L, M}⊥⊥
= s + 1 and {L, M}⊥⊥ ⊂ T . A spread is called a spread of symmetry if there is an automorphism group of Q (called the “associated group”) fixing T linewise and acting sharply transitively on the points of any of its lines. One easily sees that a spread of symmetry is necessarily a regular spread. The following theorem allows one to reverse the construction of Payne. Theorem 1.2 (Theorem 2.7 of M. De Soete and J. A. Thas [1]) Let P be a GQ of order (s − 1, s + 1) with a spread of symmetry T . Then P can be obtained by Payne derivation from a GQ of order s with a center of symmetry. 1 In the literature these spreads are also called normal or Hermitian spreads.

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In fact, the existence of a regular spread already is sufficient to reverse Payne’s construction. Theorem 1.3 Let P be a GQ of order (s − 1, s + 1) with a regular spread T . Then P can be obtained by Payne derivation from a GQ of order s with a regular point. The proof of this latter theorem is implicitly contained in [6] and [1]. However, as the final step in the proof of our main theorem uses this construction, we sketch it here. Let L be any line of P not in T . Then the lines of T intersecting L, together with the lines of P intersecting at least two of these lines, form an (s × s)-grid, GL 1 . Through any point of L : one line of T and one other P not in GL there are exactly two lines of P not intersecting G 1 1 L line. Using this second line and applying the above we obtain a grid GL 2 disjoint from G1 . Continuing in this way we partition the point set of P into a set GL of s disjoint (s × s)-grids L L GL 1 , G2 , . . . , Gs . This partition is easily seen to be independent of the choice of the line L L L  ∈ T in any of the grids GL 1 , G2 , . . . , Gs . Hence the relation ∼ defined by L ∼ M if and only if GL = GM with L, M  ∈ T is an equivalence relation with s + 1 equivalence classes, with each of which corresponds a set of s grids partitioning the point set. Denote these s + 1 grid sets by G0 , G1 , . . . , Gs . Define the following point-line incidence geometry Q. The points of Q are of three types: (i) the points of P , (ii) the elements (grids) of all Gi , i = 0, 1, . . . , s, (iii) a symbol (∞). The lines of Q are of two types: (a) the lines of Q not in T , (b) the s + 1 sets G0 , G1 , . . . , Gs . Finally, a point of type (i) is incident with a line of type (a) if it is incident with it in P and is never incident with a line of type (b); a point of type (ii) is incident with all lines of type (a) contained in it and with the unique line of type (b) containing it; the point (∞) is incident with no line of type (i) and with all lines of type (ii). It is easily seen that Q is a GQ of order s having (∞) as a regular point and that P is the GQ obtained by Payne derivation from Q with respect to (∞). A GQ Q is called an elation GQ (EGQ) (Qx , G) if there exists a point x of Q and a group of automorphisms G of Q such that G fixes all lines through p and acts regularly on the points not collinear with p. An EGQ (Qx , G) is called a skew translation GQ (STGQ) provided the point x is a center of symmetry and G contains the full group of symmetries about x. The classical GQ W (q) provides an example of an STGQ. Finally, the following result will turn out to be rather useful. Theorem 1.4 Let G be a p-group acting sharply transitively on the point set of a GQ of order (s, t). Then t = s + 2. Proof From |G| = (s+1)(st +1) = p k , for some natural number k, it follows that s+1 = p l , for some natural number l. Furthermore, as st + 1 > s + 1, we see that s + 1 divides st + 1, and hence s + 1 divides t − 1. If s + 1 = t − 1, we obtain the desired result. So assume by way of contradiction that t − 1 > s + 1. Then t = p l f + 1, for some natural number f > 1, and consequently |G| = (s + 1)(st + 1) = p 2l (p l f − f + 1). It follows that f is at least 1 + p l , that is, f > s + 1, and t > (s + 1)2 , contradicting Higman’s inequality t ≤ s 2 (cf. 1.2.3 of [8]).



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2 Heisenberg groups The (general) Heisenberg group Hn of dimension 2n + 1 over GF(q), with n a natural number, is the group of square (n + 2) × (n + 2)-matrices with entries in GF(q), of the following form (and with the usual matrix multiplication): ⎛ ⎞ 1 α c ⎝ 0 In β T ⎠ , 0 0 1 where α, β ∈ GF(q)n , c ∈ GF(q) and with In being the n × n-unit matrix. Let α, α  , β, β  ∈ GF(q)n and c, c ∈ GF(q); then ⎞ ⎛ ⎛ ⎞ ⎛ ⎞ 1 α  c 1 α c 1 α + α  c + c + α, β  ⎝ 0 In β T ⎠ × ⎝ 0 In β  T ⎠ = ⎝ 0 In ⎠. β + β 0 0 1 0 0 1 0 0 1 Here x, y , with x = (x1 , x2 , . . . , xn ) and y = (y1 , y2 , . . . , yn ) elements of GF(q)n , denotes x1 y1 + x2 y2 + · · · + xn yn = xy T . Thus the group Hn is isomorphic to the group {(α, c, β)  α, β ∈ GF(q)n , c ∈ GF(q)}, where the group operation ◦ is given by (α, c, β)◦(α  , c , β  ) = (α + α  , c + c + αβ  T , β + β  ). The group Hn is a p-group of order q 2n+1 , with elementary abelian center Z(Hn ) of order q, and Hn /Z(Hn ) elementary abelian of order q 2n . It is well-known, see for instance [4,7], that starting from a Heisenberg group H = H1 of dimension 3 over GF(q) with q odd, an EGQ (Qx , H) of order q can be constructed using a so-called Kantor family. As a specific example it is possible to obtain the classical GQ W (q) in this way when q is odd. In [9] various properties of Heisenberg groups are investigated related to GQs.

3 Main theorem In this section, we want to obtain the following general result: Theorem 3.1 Let Q be a GQ of order (s, √t) admitting a Singer group G, where G is a p-group and p is odd. Suppose |Z(G)| ≥ 3 |G|. Then the following properties hold. (1) We have t = s + 2, and there is a GQ Q of order s + 1 with a regular point x, such that Q is the Payne derivative of Q with respect to x. The GQ Q is an STGQ of type (Q x , K), with K isomorphic to G. √ (2) We have |Z(G)| = 3 |G|, that is, |Z(G)| = s + 1. From Theorem 3.1 the next result follows (taking the properties of Sect. 2 into account). Corollary 3.2 Let Q be a GQ of order (s, t) admitting a Singer group G, where G is a Heisenberg group of dimension 3 over the field GF(q) with q odd. Then we have the conclusion of Theorem 3.1.
We now prove Theorem 3.1. First of all, we note that t = s + 2 by Theorem 1.4, so that (s + 1)(st + 1) = (s + 1)3 . Let α be any element of G. Suppose that no point of S is mapped by α onto a point collinear with it. Then applying Benson’s theorem for automorphisms of GQs (see Theorem 1.9.1 of [8] for a precise statement), we obtain (s + 1)2 ≡ 0

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a contradiction as s + 1 is odd (as s + 1 divides the order of G). So there are points x for which x α ∼ x. We now specialize and consider elements of Z(G). Take α ∈ Z(G)× , and let x be a point for which x α ∼ x. Put L = xx α . Let y be a point incident with L and different from x, x α . Furthermore, let β ∈ G be the element that maps x onto y. Then we have x ∼ x β ⇒ x α ∼ x βα = x αβ ; x ∼ x α ⇒ x β ∼ x αβ . Hence x, x α , x β , x αβ are all on the same line, and {x, x β }α = {x α , x αβ } implies that L is fixed by α. Since G acts transitively on the point set of S , one concludes that each point is incident with a unique line that is fixed by α, that is, α fixes a spread T (α) linewise. Suppose α  ∈ Z(G)× for which T (α  )  = T (α). Then there exists a line M ∈ T (α) such that each point of M is incident with a line of T (α  ) different from M. Take any point zI M.   Then z ∼ zα I M, zα ∼ zαα  I M, so that z  ∼ zαα . On the other hand, αα  ∈ Z(G), while each element of Z(G) maps a point onto a collinear point, contradiction. So T (α) = T (α  ) and hence each element of Z(G) fixes T (α) linewise. By the regularity of G this implies that |Z(G)| ≤ s + 1. Consequently |Z(G)| = s + 1 , and T (α) is a spread of symmetry. Now (2) follows. By Theorem 1.2 it follows that Q can be constructed by Payne deriving a GQ Q with respect to a center of symmetry x. Further it is not hard to see, using the construction mentioned after Theorem 1.3, that G induces a group of automorphisms K ∼ = G of Q fixing all lines on x and acting regularly on the points not collinear with x. Now also (1) follows.


4 Concluding remarks Although Theorem 3.1 does not assume the group G to be a Heisenberg group the following conjecture seems to be reasonable. Conjecture Let G be a Singer p-group, p odd, of a GQ with the property that |Z(G)| ≥ √ 3 |G|. Then G is isomorphic to a Heisenberg group of dimension 3 over GF(q), where q is a power of p. The second author announces to be close to a proof of this conjecture. For even order Heisenberg groups of dimension 3 over GF(q) Benson’s theorem does not seem to help to obtain the desired spread of symmetry. However the authors believe that an even order Heisenberg group cannot occur as the Singer group of a GQ. Based on these results and the results of [2] we make the following conjecture, the proof of which can be seen as the final goal of the project mentioned in the introduction. Conjecture If Q is a GQ admitting a Singer group G, then there exists an EGQ (Qx , G ) of order s such that Q can be obtained from Q by Payne derivation with respect to x, and such that G ∼ = G . Acknowledgments S. De Winter and K. Thas are Postdoctoral Fellows of the Fund for Scientific Research — Flanders (Belgium).

References 1. De Soete M., Thas J.A.: A coordinatization of generalized quadrangles of order (s, s + 2). J. Combin. Theory Ser. A 48, 1–11 (1988).

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2. De Winter S., Thas K.: Generalized quadrangles with an abelian Singer group. Des. Codes Cryptogr. 39, 81–87 (2006). 3. Ghinelli D.: Regular groups on generalized quadrangles and nonabelian difference sets with multiplier −1. Geom. Dedicata 41, 165–174 (1992). 4. Kantor W.M.: Generalized polygons, SCABS and GABS. In Lecture Notes in Mathematics (Buildings and the Geometry of Diagrams, Como, 1984), Springer-Verlag, 79–158. (1986). 5. Payne S.E.: Non-isomorphic generalized quadrangles. J. Algebra 18, 201–212 (1971). 6. Payne S.E.: Quadrangles of order (s − 1, s + 1). J. Algebra 22, 97–119 (1972). 7. Payne S.E.: Finite groups that admit Kantor families. In: Hulpke, Liebler, Penttila, Seress Finite Geometries (eds.) Groups and Computation. Proceedings of the conference Finite Geometries, Groups, and Computation, September 4–9, 2004, Pingree Park, Colorado, USA, pp. 191–202 (2006). 8. Payne S.E., Thas J.A.: Finite Generalized Quadrangles. Research Notes in Mathematics 110, Pitman Advanced Publishing Program, Boston/London/Melbourne, 1984. 9. Thas K.: Lectures on Elation Quadrangles, 100 pp (Preprint).

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