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### A Characterization ofQ(5, q) Using One SubquadrangleQ(4,q )

Leen Brouns, Joseph A. Thas, Hendrik van Maldeghem
<span title="">2002</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/54t3hgai4fhhthc74mj7z7tapu" style="color: black;">European journal of combinatorics (Print)</a> </i> &nbsp;
Further, for q odd, we prove that if every triad {x, y, z} of is 3-regular in and {x, y, z} ⊥⊥ ⊂ , then is classical.  ...  in PG(5, q), denoted by Q(5, q) and of order (q, q 2 ); it is the dual of H (3, q 2 );  ...  As n 2 and n 3 are on lines of {L , K } ⊥ , both conics C n 2 := O ∩ O n 2 and C n 3 := O ∩ O n 3 belong to the flock F of O.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1006/eujc.2001.0554">doi:10.1006/eujc.2001.0554</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vaiy4j5bobfh3a32ydyuvywp4a">fatcat:vaiy4j5bobfh3a32ydyuvywp4a</a> </span>
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S.G. Barwick, Matthew R. Brown, Tim Penttila
<span title="">2006</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/z77xaqun7bcxjkh75wb7iseaty" style="color: black;">Journal of combinatorial theory. Series A</a> </i> &nbsp;
It was shown by Thas in 1987 that to such flocks correspond generalized quadrangles of order (q 2 , q), previously constructed algebraically by Kantor (q odd) and Payne (q even).  ...  A flock of a quadratic cone of PG(3, q) is a partition of the non-vertex points into plane sections.  ...  Each element of * contains q 2 conics containing * ∞ , and each such conic is contained in exactly q elements of * by Lemma 17. Hence there are q 4 (q − 1) such conics.  ...
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### Some remarks on flocks

Laura Bader, Christine M. O'Keefe, Tim Penttila
<span title="">2004</span> <i title="Cambridge University Press (CUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/qucxg3yndjaptj6krclryhveua" style="color: black;">Journal of the Australian Mathematical Society</a> </i> &nbsp;
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 9) .  ...  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?  ...  Aknowledgements This work was done partly while the first author was a Visiting Professor at the University of Adelaide and at the University of Western Australia, partly while the second author was a  ...
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### Subquadrangles of order s of generalized quadrangles of order (s,s2), Part II

M.R. Brown, J.A. Thas
<span title="">2004</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/z77xaqun7bcxjkh75wb7iseaty" style="color: black;">Journal of combinatorial theory. Series A</a> </i> &nbsp;
Finally, it is shown that if S is a flock GQ of order ðs 2 ; sÞ; s odd, with base point ðNÞ and S 0 is a subquadrangle of order s of S containing the point ðNÞ; then S is a Kantor semifield flock GQ, S  ...  there is a characterization of the Kantor semifield flock GQ in terms of the net defined by the regular point ðNÞ of the flock GQ. r  ...  The GQ SðF Þ is the classical GQ Hð3; q 2 Þ if and only if the flock F is linear, that is, if and only if all planes of the conics of the flock contain a common line.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jcta.2003.12.007">doi:10.1016/j.jcta.2003.12.007</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gxnqf52vprg2jemf4o3gim42hu">fatcat:gxnqf52vprg2jemf4o3gim42hu</a> </span>
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### Page 5620 of Mathematical Reviews Vol. , Issue 2001H [page]

<span title="">2001</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
, Tim (5-WA; Nedlands) Triads, flocks of conics and Q~ (5,q).  ...  The authors prove that any ovoid of Q(4,q), g even, admitting a flock of conics (i.e., a partition into two conics and two points) is an elliptic quadric Q~ (3,q).  ...
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### Subquadrangles of order s of generalized quadrangles of order (s,s2), Part I

M.R. Brown, J.A. Thas
<span title="">2004</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/z77xaqun7bcxjkh75wb7iseaty" style="color: black;">Journal of combinatorial theory. Series A</a> </i> &nbsp;
In the case where S is a dual flock GQ of order ðs; s 2 Þ; s even, the classification of subquadrangles of order s is completed.  ...  In the case where E is an egg good at an element and the corresponding translation generalized quadrangle TðEÞ of order ðs; s 2 Þ; s even, has base point ðNÞ it is proved that if TðEÞ has a subquadrangle  ...  to a conic by adding X and Y ).  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jcta.2003.12.006">doi:10.1016/j.jcta.2003.12.006</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/r7g6d5ahqzcbpfurrchbthvzbe">fatcat:r7g6d5ahqzcbpfurrchbthvzbe</a> </span>
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### Flocks of Cones: Star Flocks [article]

William Cherowitzo
<span title="2009-11-04">2009</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The concept of a flock of a quadratic cone is generalized to arbitrary cones. Flocks whose planes contain a common point are called star flocks.  ...  This connection can also be used to obtain examples of bilinear flocks of non-quadratic cones.  ...  The analogous example for even q is the projective triad of side (q + 2)/2 (see [19] ).  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0911.0725v1">arXiv:0911.0725v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6yqzg627lff4xn5xjub436e7iy">fatcat:6yqzg627lff4xn5xjub436e7iy</a> </span>
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### Isomorphisms of groups related to flocks

Koen Thas
<span title="2011-11-12">2011</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/cvausbtygzb5fdr6mvjzyvlh6e" style="color: black;">Journal of Algebraic Combinatorics</a> </i> &nbsp;
Denote the class of such flock quadrangles by C . In this paper, we resolve the question of Payne for the complete class C .  ...  A truly fruitful way to construct finite generalized quadrangles is through the detection of Kantor families in the general 5-dimensional Heisenberg group H 2 (q) over some finite field F q .  ...  Payne, Hendrik Van Maldeghem and the referees for several useful suggestions. The author is partly supported by the Fund for Scientific Research-Flanders (Belgium).  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10801-011-0326-0">doi:10.1007/s10801-011-0326-0</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wvmusbupmza7xafqipcrz6zbue">fatcat:wvmusbupmza7xafqipcrz6zbue</a> </span>
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### Generalized quadrangles of orrder (s, s2), I

J.A Thas
<span title="">1994</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/z77xaqun7bcxjkh75wb7iseaty" style="color: black;">Journal of combinatorial theory. Series A</a> </i> &nbsp;
Also, as a by-product, several classes of ovoids of Q(4, s), s odd, are obtained; one of these classes is new.  ...  It was shown by Payne that each generalized quadrangle of order (s 2, s), s > 1, arising from a flock of a quadratic cone, has property (G) at its point (~).  ...  of K-{x} into q disjoint irreducible conics.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0097-3165(94)90009-4">doi:10.1016/0097-3165(94)90009-4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/febemdkbnvei5mmgysv5eb4cxq">fatcat:febemdkbnvei5mmgysv5eb4cxq</a> </span>
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### On generalized quadrangles with some concurrent axes of symmetry

Koen Thas
<span title="">2002</span> <i title="The Belgian Mathematical Society"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ogod2hdpqfhkza4lug2fpzfqae" style="color: black;">Bulletin of the Belgian Mathematical Society Simon Stevin</a> </i> &nbsp;
Let S be a finite Generalized Quadrangle (GQ) of order (s, t), s = 1 = t, and suppose L is a line of S. A symmetry about L is an automorphism of S which fixes every line concurrent with L.  ...  A line L is an axis of symmetry if there is a full group of size s of symmetries about L. A point of a generalized quadrangle is a translation point if every line through it is an axis of symmetry.  ...  The author is a research fellow supported by the Flemish Institute for the promotion of Scientific and Technological Research in Industry (IWT), grant no. IWT/SB/991254/Thas.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.36045/bbms/1102715101">doi:10.36045/bbms/1102715101</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/iyeoqlhl3bf57hn4ysowpvuvma">fatcat:iyeoqlhl3bf57hn4ysowpvuvma</a> </span>
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### Generalized quadrangles from a local point of view

Koen Thas
<span title="">2011</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/bupqdehysjdt7p33mveughlzz4" style="color: black;">Journal of Geometry</a> </i> &nbsp;
In this lecture, I will survey several recent results in the local theory of generalized quadrangles.  ...  Flock quadrangles and q-clans Let F be a flock of the quadratic cone K in PG(3, q) with equation X 0 X 1 = X 2 2 ; so F is a partition of K without the vertex consisting of irreducible conics.  ...  We finally mention that if S (F ) is a flock quadrangle of order (q 2 , q), and its dual is a TGQ, then S (F ) ∼ = H(3, q 2 ) if q is even.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00022-011-0092-0">doi:10.1007/s00022-011-0092-0</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/e3jhkhhlxrhqxinhif2wbd7yyq">fatcat:e3jhkhhlxrhqxinhif2wbd7yyq</a> </span>
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J. A. Thas, H. Van Maldeghem
<span title="">2006</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/uthuanwbp5dnbly3kjw3ltqx4a" style="color: black;">Journal of Mathematical Sciences</a> </i> &nbsp;
Flocks and generalized quadrangles Let F be a flock of the quadratic cone K with vertex x of PG (3, q) , that is, a partition of K\{x} into disjoint irreducible conics.  ...  The set of these q + 1 spaces will be called an inflated π 0 -conic.  ...
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### TRANSLATION GENERALIZED QUADRANGLES FOR WHICH THE TRANSLATION DUAL ARISES FROM A FLOCK

KOEN THAS
<span title="">2003</span> <i title="Cambridge University Press (CUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/bmxdrsjtrrgljhk5nwkdtjefke" style="color: black;">Glasgow Mathematical Journal</a> </i> &nbsp;
, q), subtended ovoids in generalized quadrangles, and the understanding of automorphism groups of certain generalized quadrangles.  ...  There are important consequences for the theory of generalized ovoids (or eggs) in PG(4n − 1, q), the study of span-symmetric generalized quadrangles, derivation of flocks of the quadratic cone in PG(3  ...  The author is a research fellow supported by the Flemish Institute for the promotion of Scientific and Technological Research in Industry (IWT), grant no. IWT/SB/991254/Thas.  ...
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### Eggs in PG(4n−1,q), q even, containing a pseudo-pointed conic

Matthew R. Brown, Michel Lavrauw
<span title="">2005</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/54t3hgai4fhhthc74mj7z7tapu" style="color: black;">European journal of combinatorics (Print)</a> </i> &nbsp;
We prove that an egg in PG(4n − 1, q), q even, contains a pseudo pointed conic, that is, a pseudo-oval arising from a pointed conic of PG(2, q n ), q even, if and only if the egg is elementary and the  ...  Brown ([5]) proved that if an ovoid of PG(3, q), q even, contains a pointed conic, then either q = 4 and the ovoid is an elliptic quadric, or q = 8 and the ovoid is a Tits ovoid.  ...  By embedding PG(2, q n ) in PG(3, q n ) and dualising in PG(3, q n ) one sees that the set of affine points of any n-space intersecting PG(2, q n ) in U becomes a set of planes forming a semifield flock  ...
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### Classification of skew translation generalized quadrangles, I

Koen Thas
<span title="2015-02-12">2015</span> <i title="Centre pour la Communication Scientifique Directe (CCSD)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/aagtqr2vajamvduhte7kigeygi" style="color: black;">Discrete Mathematics &amp; Theoretical Computer Science</a> </i> &nbsp;
Some of these involve, and solve, long-standing open problems.  ...  Combinatorics International audience We describe new classification results in the theory of generalized quadrangles (= Tits-buildings of rank 2 and type B2), more precisely in the (large) subtheory of  ...  A flock is a partition of the F q -rational points of K \ {vertex} into q disjoint irreducible conics.  ...
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