A coordinatization of generalized quadrangles of order (s, s + 2)

Soete, Marijke De; Thas, J. A.

doi:10.1016/0097-3165(88)90070-2

Cited by 10 publications

(10 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Some of these results have already been obtained by De Soete and Thas [4], who define a (0, 2)-set in 5? to be a non-empty set SB of pairwise non-collinear points such that IX 1 n 38\ e {0, 2} for all X e & \ B §.…”

Section: Lemma 22 Let S3 Be a Partial Blt-sef In The Generalised Qumentioning

confidence: 66%

“…Here we consider a number of generalisations of BLT-sets of Q(4, q), q odd: to BLT-sets of H (3, q 2 ), q odd, in Section 5; to BLT-sets of T*(^), in Section 6; to BLT-sets of finite generalised quadrangles in general, in Section 2; to BLT-sets of the polar spaces Q(2n, q), q odd, of rank n > 2, in Section 7. The material in Section 2 is related to work of De Soete and Thas [4], predating the introduction of BLT-sets of Q(4, q) in Bader, Lunardon and Thas [2] by six years. While we note the advances of Shult and Thas [14] in Section 3, we restrict ourselves to BLT-sets of points, rather than sets of subspaces with the BLT-property.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Some remarks on flocks

Bader¹,

O'Keefe

Penttila³

2004

J. Aust. Math. Soc.

View full text Add to dashboard Cite

New proofs are given of the fundamental results of Bader, Lunardon and Thas relating flocks of the quadratic cone in PG(3, q), q odd, and BLT-sets of Q(4, q). We also show that there is a unique BLT-set of H(3, 9). The model of Penttila for Q(4, q), q odd, is extended to Q(2m, q) to construct partial flocks of size qm/2 + m/2 -1 of the cone Jt in PG(2m -1, q) with vertex a point and base Q(2m -2,q), where q is congruent to 1 or 3 modulo 8 and m is even. These partial flocks are larger than the largest previously known for m > 2. Also, the example of O'Keefe and Thas of a partial flock of Jf in PG(5, 3) of size 6 is generalised to a partial flock of the cone Jt? of PG(2pn -1, p) of size 2pn, for any prime p congruent to 1 or 3 modulo 8, with the corresponding partial BLT-set of Q(2pn, p) admitting the symmetric group of degree 2pn + 1.2000 Mathematics subject classification: primary 51E12; secondary 51A40, 51A5O, 51E20, O5B25.

show abstract

Section: Lemma 22 Let S3 Be a Partial Blt-sef In The Generalised Qumentioning

confidence: 66%

Section: Introductionmentioning

confidence: 99%

Some remarks on flocks

Bader¹,

O'Keefe

Penttila³

2004

J. Aust. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…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.…”

Section: Theorem 13 Let P Be a Gq Of Order (S − 1 S + 1) With A Regmentioning

confidence: 96%

Generalized quadrangles admitting a sharply transitive Heisenberg group

Winter

Thas

2007

Des. Codes Cryptogr.

View full text Add to dashboard Cite

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.

show abstract

“…The case where S is a spread of symmetry of a GQ Q of order (s, s + 2) has already been considered in [11].…”

Section: The Case T = S +mentioning

confidence: 99%

Generalized Quadrangles with a Spread of Symmetry

Bruyn

1999

European Journal of Combinatorics

View full text Add to dashboard Cite

We present a common construction for some known infinite classes of generalized quadrangles. Whether this construction yields other (unknown) generalized quadrangles is an open problem. The class of generalized quadrangles obtained this way is characterized in two different ways. On the one hand, they are exactly the generalized quadrangles having a spread of symmetry. On the other hand, they can be characterized in terms of the group of projectivities with respect to a spread. We explore some properties of these generalized quadrangles. All these results can be applied to the theory of the glued near hexagons, a class of near hexagons introduced by the author in De Bruyn (1998) On near hexagons and spreads of generalized quadrangles, preprint.

show abstract

A coordinatization of generalized quadrangles of order (s, s + 2)

Cited by 10 publications

References 5 publications

Some remarks on flocks

Some remarks on flocks

Generalized quadrangles admitting a sharply transitive Heisenberg group

Generalized Quadrangles with a Spread of Symmetry

Contact Info

Product

Resources

About