Elation generalized quadrangles with extra automorphisms and trivial spans

We show that the parameters of a finite elation generalized quadrangle for which the point spans containing the elation point (∞) are trivial, and which admit extra automorphisms fixing (∞) linewise, are powers of the same prime, so that the elation group is a p-group for some prime p. This observation has several strong corollaries, perhaps the most remarkable of which is that a thick elation quadrangle of order (s, s2) (for the natural number s) that admits an extra automorphism of the aforementioned type which is an involution, is a translation generalized quadrangle—that is, the elation group is elementary abelian. c © 2008 Published by Elsevier Ltd 1. Generalized quadrangles and regularity We assume that the reader is familiar with the basic combinatorial notions from the theory of generalized quadrangles. Details can be found in the monograph [5]. We only recall the concept of regularity in a GQ of order (s, t) with point set P . For p ∈ P , put p = {q ∈ P ‖ q ∼ p} (and note that p ∈ p). For a pair of distinct points {p, q}, we denote p∩q also by {p, q}. Then |{p, q}| = s+1 or t +1, according as p ∼ q or p q , respectively. For p 6= q , we define {p, q} = {r ∈ P ‖ r ∈ s for all s ∈ {p, q}}. We have that |{p, q}| = s+1 or |{p, q}| ≤ t+1 according as p ∼ q or p q, respectively. If p ∼ q , p 6= q , or if p q and |{p, q}| = t + 1, we say that the pair {p, q} is regular. The point p is regular provided {p, q} is regular for every q ∈ P\{p}. One easily proves that either s = 1 or t ≤ s if S has a regular pair of non-collinear points; see [5, 1.3.6]. E-mail address: [email protected]. 0195-6698/$ see front matter c © 2008 Published by Elsevier Ltd doi:10.1016/j.ejc.2007.02.011 440 K. Thas / European Journal of Combinatorics 29 (2008) 439–442 2. Elation generalized quadrangles Let S = (P, B, I) be a GQ. An automorphism of S is a permutation of P ∪ B that fixes P , B, and preserves the incidence relation. The set of all automorphisms of S naturally forms a group, denoted as Aut(S), and any subgroup of Aut(S) is called “automorphism group” of S. Let A ⊆ P , and H ≤ Aut(S). Then H[A] denotes the subgroup of H of elements that fix all elements of B that are incident with the points of A. A thick generalized quadrangle which admits an automorphism group G that fixes some point (∞) linewise while acting sharply transitively on the set of points not collinear with (∞) is called an elation generalized quadrangle (EGQ), and (∞) is an elation point. The group G is an elation group. We often write (S,G) or S when we explicitly want to indicate what the elation point and group or elation point, respectively, is. If an elation group of an EGQ is abelian, we call it a translation generalized quadrangle (TGQ); the elation point and elation group are then usually called the “translation point” and “translation group”. The theory of EGQs is extremely deep and important. The reason for this can already be read from the following observation: Each known generalized quadrangle is, up to duality, an EGQ, or can be constructed from one by “Payne derivation”. Even for EGQs with an abelian elation group, there exists a vast literature—cf. the recent monograph [6]. The following, which we state in geometrical terms, is a corollary of a result of Hachenberger [4] that solved an important conjecture of S.E. Payne. The proof of the equivalence of this geometrical statement and the formulation of Hachenberger is not trivial, and can be found in the monograph [7, Chapter 3]. Theorem 2.1 ([4, Hachenberger]). Let (S,G) be a thick EGQ of order (s, t), and suppose (∞) is a regular point. Then G is a p-group for some prime p, and st is a power of p. More general versions of Theorem 2.1 can be found in [4], and the recent paper [10] (in which a conjecture stated in the first reference was solved). Remark 2.2. Theorem 2.1 was independently obtained by Chen, see [1], who proved that if S is a thick EGQ of order (s, t) and if there is a point y (∞) for which |{y, (∞)}| ≥ √ t + 1, one has the same conclusion. One can prove that, with essentially the same technique that Chen uses, this result can be strengthened a bit to |{y, (∞)}| ≥ 4 √ t + 1; see [8]. Theorem 2.1 is one of the fundamental achievements in the theory around Kantor’s conjecture: Kantor’s Conjecture. The elation group of a finite thick EGQ is a p-group for some prime p. Theorem 2.1 considers EGQs for which the spans containing (∞) are theoretically maximal. Here, we consider the other side of the spectrum; we want to look at elation quadrangles for which the spans containing (∞) are trivial. In comparison with the situation sketched in Theorem 2.1, there is quite some loss of geometrical information— therefore, we invoke a further assumption, namely that there are elements in Aut(S)[(∞)]\G. The advantage of this assumption is that we can say more than just stating that G is a p-group; in some cases the structure of the elation group is partly revealed. The proof of the main result presented in this note depends heavily on two fundamental developments on Kantor’s Conjecture: Frohardt’s result [2] and the recent solution by the author of conjectures made by Knarr [9] and Hachenberger [4]. Recall that Frohardt’s result is the following: K. Thas / European Journal of Combinatorics 29 (2008) 439–442 441 Theorem 2.3 ([2, Frohardt]). Let S be a thick EGQ of order (s, t) with s ≤ t or with nilpotent elation group. Then st is a prime power. 3. A structure theorem on subGQs A subquadrangle, or also subGQ, S ′ = (P , B , I) of a GQ S = (P, B, I) is a GQ for which P ′ ⊆ P, B ′ ⊆ B, and where I is the restriction of I to (P ′ × B ) ∪ (B ′ × P ). Let (S(x),G) be a thick EGQ of order (s, t), and let S ′ be a thick subGQ of order (s, t) containing x . Then the following result was observed in [10]; see also [9]. Theorem 3.1 ([10, Thas]). If the point x is not regular, G is a p-group for some prime p and s = st . We collect this theorem and some other recent results in the following theorem. Call for that purpose a subGQ of order (s, t ) full if either s = s or t ′ = t . Theorem 3.2. Let (S(x),G) be a thick EGQ of order (s, t), and let S ′ be a full subGQ of order (s, t ) containing x with (s, t ) 6= (s, t). Then G is a p-group for some prime p and s and t are powers of p. Proof. First suppose S ′ is not thick. If s = 1 and t ′ = t , then x is regular, and the result can be found in [7]. Let t ′ = 1 and s = s; then S contains some regular line pair, so s ≤ t . The result follows from Theorem 2.3. We may suppose S ′ is thick, and w.l.o.g. that no full thin subGQs contain x . If t ′ = t , then Theorem 3.1 yields the desired observation. If s = s, [5, Section 2.2.2] leads to s ≤ t , and again Frohardt’s Theorem helps us to conclude that G is a p-group for some prime p. 4. Elation quadrangles with extra automorphisms fixing (∞) linewise, and trivial spans Let (S,G) be a thick EGQ of order (s, t), and suppose {(∞), z} = {(∞), z} for at least one, and then any point, z ∈ P\(∞) (P is the point set of S). Suppose furthermore that Aut(S)[(∞)] 6= G. Let φ ∈ Aut(S)[(∞)]\G, and define G = 〈G, φ〉. Then G is a transitive but not sharply transitive group in its action on P\(∞) =: X . Suppose some element α 6= 1 in G fixes at least two points of X . Then since these points cannot be in the same span containing (∞), the fixed elements structure of α is a thick subGQ of order (s, t); cf. [5, Section 8.1.1]. By Theorem 3.1 we conclude that G is a p-group for some prime p. So now we suppose that only 1 fixes two distinct points in X . This means that (G, X) is a Frobenius group, so the Frobenius kernel K ≡ G E G acts sharply transitively on X . Furthermore, K is known to be nilpotent (see, e.g., [3, p. 339]). By Theorem 2.3, we conclude that G is a p-group for some prime p. Moreover, if the Frobenius complement G∗z has even size, then K = G is abelian [3], so that G is elementary abelian by [5, Chapter 8]. In this case (S,G) is a translation generalized quadrangle, and s ≤ t ; see [5, Chapter 8]. We have proved Theorem 4.1: Theorem 4.1. Let (S,G) be a thick EGQ of order (s, t), and suppose {(∞), z} = {(∞), z} for at least one point z not collinear with (∞). Suppose that Aut(S)[(∞)], the full group of automorphisms of S that fix (∞) linewise, does not coincide with G. Then G is a p-group for the prime p, and s and t are powers of p. The proof has the following application for TGQs. 442 K. Thas / European Journal of Combinatorics 29 (2008) 439–442 Corollary 4.2. Let (S,G) = (P, B, I) be a thick EGQ of order (s, t). Suppose that Aut(S)[(∞)] = W does not equal G, and that {(∞), z} = {(∞), z} for at least one point z not collinear with (∞). (a) If (W, P\(∞)) is a Frobenius group and the Frobenius complement has even size, then G is an elementary abelian p-group for some odd prime p, and S is a TGQ. (b) If S has no subGQ of order (s, t) containing (∞), then G is a p-group for some prime p. In particular, if s ≤ t , (b) applies. (c) If S has no subGQ of order (s, t) containing (∞) and |W | is even while |G| is odd, then we have the same conclusion as in (a). Again the observation applies when one demands s ≤ t instead of the first assumption of (c). Any TGQ (of appropriate order) satisfies the conditions of the latter theorem. Note that if, in the two preceding results, t = s2, then automatically all spans of type {(∞), z} have size 2 by [5, Section 1.4.2(ii)], and no subGQs of order (s, t) can occur by [5, Section 2.2.2]. So the reader can think up the consequences for this particular case. We only state: Theorem 4.3. Let (S,G) = (P, B, I) be a thick EGQ of order (s, s2). Suppose that [Aut(S)[(∞)] : G] is even. Then G is elementary abelian, so that S is a TGQ. Remark 4.4. Theorem 4.3 seems to indicate that “many” EGQs (w.r.t. points) of order (s, s2) are TGQs. We end with a characterization of semifield flock GQs. For standard notation and results on the latter objects, we refer the reader to the monograph [6]. If S(F) is a non-classical flock GQ of order (q2, q) and its point–line dual is a TGQ, then S(F) contains an elation line incident with (∞). Theorem 4.5. Let S(F) be a flock GQ of order (q2, q), and suppose S(F)D is an EGQ. Let MI(∞) be the elation line of S(F) that is incident with the special point (∞). If there is a line N M such that |Aut(S(F))[L ,M]| is even, then S(F)D is a TGQ with translation point M, so that F is a semifield flock up to derivation. When q is even, the TGQ is of classical type.