Regular groups on generalized quadrangles and nonabelian difference sets with multiplier -1

Ghinelli, Dina

doi:10.1007/bf00182417

Cited by 23 publications

(28 citation statements)

References 9 publications

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“…Remark 1.3. GAP [10] computations show that the pseudo-hyperoval of PG (11,2) obtained by field reduction of the Lunelli-Sce hyperoval of PG(2, 16) has a stabilizer with 14 transitive subgroups, with half of these subgroups fixing a 5-dimensional projective subspace. Details are given in the Appendix.…”

Section: Introduction and Resultsmentioning

confidence: 99%

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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Section: Introduction and Resultsmentioning

confidence: 99%

Generalized Quadrangles and Transitive Pseudo‐Hyperovals

Bamberg

Glasby

Popiel

et al. 2014

J of Combinatorial Designs

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“…It is important to notice that there exists a very interesting example of a PG-regulus R in PG(5, 3) yielding a pg (8,20,2). This PG-regulus is due to Mathon [5] and is the only known PG-regulus yielding a pg(s, 2(s + 2), 2).…”

Section: Pairs (S G) Of Spread-typementioning

confidence: 99%

“…This motivated Ghinelli [8] to study generalized quadrangles (GQs) admitting a (not necessarily abelian) Singer group. At this point we want to mention that looking at GQs is natural as both projective planes and GQs are members of the larger class of generalized polygons.…”

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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“…In the past 15 years, the question has been posed several times whether there are fruitful Singer group/difference set theories for other types of (building-like) geometries, especially for the other generalized n-gons. For generalized 4-gons, or also "generalized quadrangles" (see Section 2 for a formal definition), such a theory was initiated by Ghinelli [5], where it was shown, amongst other things, that a finite generalized quadrangle of order s (> 1) cannot admit an abelian Singer group.…”

Section: Introductionmentioning

confidence: 99%