Generalized Quadrangles with an Abelian Singer Group

Winter, Stefaan De; Thas, Koen

doi:10.1007/s10623-005-2747-z

Cited by 13 publications

(38 citation statements)

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“…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.…”

Section: Discussionmentioning

confidence: 85%

“…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.…”

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confidence: 99%

“…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]).…”

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confidence: 99%

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Generalized quadrangles admitting a sharply transitive Heisenberg group

Winter

Thas

2007

Des. Codes Cryptogr.

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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.AMS Classifications 05B10 · 05B25 · 05E20 · 20B25 · 51E12 · 51E20 Singer generalized quadranglesA 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.

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Section: Discussionmentioning

confidence: 85%

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confidence: 99%

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confidence: 99%

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Generalized quadrangles admitting a sharply transitive Heisenberg group

Winter

Thas

2007

Des. Codes Cryptogr.

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“…In [7] it is shown that for every partial geometry S with α = 1, that is every finite generalized quadrangle, which admits an abelian Singer group G, the pair (S, G) is of spread-type, yielding that S is the generalized linear representation of a generalized hyperoval.…”

Section: Pairs (S G) Of Spread-typementioning

confidence: 99%

“…In [7] they obtained that every finite GQ which admits an abelian Singer group necessarily arises as the generalized linear representation of a generalized hyperoval. Further it was noted in that paper that no finite generalized n-gon with n > 4 can admit an abelian Singer group.…”

Section: Introductionmentioning

confidence: 99%

Partial geometries pg (s, t, 2) with an abelian Singer group and a characterization of the van Lint-Schrijver partial geometry

Winter

2006

J Algebr Comb

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Let S be a proper partial geometry pg(s, t, 2), and let G be an abelian group of automorphisms of S acting regularly on the points of S. Then either t ≡ 2 (mod s + 1) or S is a pg(5, 5, 2) isomorphic to the partial geometry of van Lint and Schrijver (Combinatorica 1 (1981), 63-73). This result is a new step towards the classification of partial geometries with an abelian Singer group and further provides an interesting characterization of the geometry of van Lint and Schrijver.

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Generalized Quadrangles and Transitive Pseudo‐Hyperovals

Bamberg

Glasby

Popiel

et al. 2014

J of Combinatorial Designs

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A pseudo-hyperoval of a projective space PG(3n − 1, q), q even, is a set of q n + 2 subspaces of dimension n − 1 such that any three span the whole space. We prove that a pseudohyperoval with an irreducible transitive stabilizer is elementary. We then deduce from this result a classification of the thick generalized quadrangles Q that admit a point-primitive, linetransitive automorphism group with a point-regular abelian normal subgroup. Specifically, we show that Q is flag-transitive and isomorphic to T * 2 (H), where H is either the regular hyperoval of PG(2, 4) or the Lunelli-Sce hyperoval of PG(2, 16). (i) H acts transitively on the lines of Q;(ii) H acts transitively on the flags of Q; (iii) H acts transitively on the pseudo-hyperoval {U 1 , U 2 , . . . , U t+1 }, by conjugation.Theorem 1.1 is proved in Section 2. It implies that a classification of transitive pseudohyperovals would yield a classification of the generalized quadrangles that admit a line-transitive automorphism group with a point-regular abelian normal subgroup, and, moreover, that such generalized quadrangles are, in fact, flag-transitive. By a result of J. A. Thas [28, §4.5], if a projective space PG(3n − 1, q) contains a pseudo-hyperoval, then q = 2 f for some positive integer f . For small values of the product nf , we appeal to some existing results to classify the transitive pseudo-hyperovals of PG(3n − 1, 2 f ), Journal of Combinatorial Designs

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Generalized Quadrangles with an Abelian Singer Group

Abstract: In this note we characterize thick finite generalized quadrangles constructed from a generalized hyperoval as those admitting an abelian Singer group, i.e., an abelian group acting regularly on the points.

Cited by 13 publications

References 6 publications

Generalized quadrangles admitting a sharply transitive Heisenberg group

Generalized quadrangles admitting a sharply transitive Heisenberg group

Partial geometries pg (s, t, 2) with an abelian Singer group and a characterization of the van Lint-Schrijver partial geometry

Generalized Quadrangles and Transitive Pseudo‐Hyperovals

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