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Ovoids of PG(3,q) stabilized by a central collineation

2003
*
European journal of combinatorics (Print)
*

In this paper we give

doi:10.1016/s0195-6698(03)00028-3
fatcat:mksray4gjrerxlfpy3f334mola
*a*short proof that an*ovoid**of**PG*(*3*,*q*) is*stabilized**by**a*non-trivial*central**collineation**of**PG*(*3*,*q*) if and only if it is an elliptic quadric. ... Acknowledgement The author acknowledges the support*of*the University*of*Ghent grant GOA 12050300. ... An*ovoid**of**PG*(*3*,*q*) is*stabilized**by**a**central**collineation*if and only if it is an elliptic quadric. ...##
###
Regular Packings of PG (3,q)

1998
*
European journal of combinatorics (Print)
*

Two regular packings

doi:10.1006/eujc.1998.0239
fatcat:nxeyvnlsm5c3nonjhmtmm75gu4
*of**PG*(*3*,*q*) are constructed whenever*q*≡ 2 (mod*3*), with each packing admitting*a*cyclic group*of*order*q*2 +*q*+ 1 acting regularly on the regular spreads in the packing. ... The resulting families*of*translation planes*of*order*q*4 include the Lorimer-Rahilly and Johnson-Walker planes*of*order 16. ... Now G acts faithfully on*PG*(*3*,*q*)/P or on π (as G S contains no nontrivial*central**collineations*), so G ∼ =Ḡ ≤*PG*L(*3*,*q*): letr be the resulting image in*PG*L(*3*,*q*)*of*r . ...##
###
Group theoretic characterizations of Buekenhout–Metz unitals in $\mathop{\mathrm{PG}}(2,q^{2})$

2010
*
Journal of Algebraic Combinatorics
*

Let G be the group

doi:10.1007/s10801-010-0250-8
fatcat:ghlhjk4ma5btzeovcsw6zftd7i
*of*projectivities*stabilizing**a*unital U in*PG*(2,*q*2 ) and let*A*, B be two distinct points*of*U. ... In this paper we prove that, if G has an elation group*of*order*q*with center*A*and*a*group*of*projectivities*stabilizing*both*A*and B*of*order*a*divisor*of**q*− 1 greater than 2( √*q*− 1), then U is an ... Next we recall that*a**central**collineation**of**PG*(2,*q*2 ) is*a**collineation*α fixing every point*of**a*line (the axis*of*α) and fixing every line through*a*point C (the center*of*α). ...##
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On the nonexistence of certain Hughes generalized quadrangles

2007
*
Designs, Codes and Cryptography
*

that every

doi:10.1007/s10623-007-9066-5
fatcat:mpg2crfalnb7pmysafkvfs7nrm
*central*elation*of*the subquadrangle is induced*by**a**collineation**of*Γ. ... In this paper, we prove that the Hermitian quadrangle H(4,*q*2 ) is the unique generalized quadrangle Γ*of*order (*q*2 ,*q**3*) containing some subquadrangle*of*order (*q*2 ,*q*) isomorphic to H(*3*,*q*2 ) such ... In particular, if g is*a**central**collineation*in ∆, then it is also*a**central**collineation*in Γ. Proof. The*stabilizer*G O acts on O as PGU*3*(*q*). ...##
###
The (56, 11, 2) design of Hall, Lane, and Wales

1973
*
Journal of combinatorial theory. Series A
*

The hint for this construction comes from the remark in [

doi:10.1016/0097-3165(73)90068-x
fatcat:3pst3d3mhjh73esg4cqhvv3shu
*3*] that the points*of*D can be looked upon as*a*set*of**ovoids*in the projective plane*PG*(2,4) upon which PSL(*3*, 4) acts. ... Using the representation*of*D derived from S" it is possible to compute that D has*a*group*of**collineations*which is isomorphic to*a*subgroup*of*index*3*in the group*of*all*collineations*and correlations ... (*3*, 4) extended*by*an element*of*order 2 which in*a*suitable coordinate system*of**PG*(2, 4) corresponds to*a*field automorfihism*of*GF(4). ...##
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Page 2223 of Mathematical Reviews Vol. , Issue 2004c
[page]

2004
*
Mathematical Reviews
*

De Clerck (B-GHNT-PM; Ghent)
2004c:51006 51E20 05B25
Brown, Matthew R. (5-ADLD-SPM; Adelaide)

*Ovoids**of**PG*(*3*, ¢)*stabilized**by**a**central**collineation*. (English summary) European J. ... An*ovoid**of**PG*(*3*,*q*) is*a*set*of*g? +1 points no three*of*which are collinear. An elliptic quadric is such an object, and the converse is true for g odd, and for g = 4,16. ...##
###
Codes from Generalized Hexagons

2005
*
Designs, Codes and Cryptography
*

These in turn are constructed using

doi:10.1007/s10623-005-4035-3
fatcat:dzvjcui3kjen7prowy4twaks7y
*a*new distance-2-*ovoid**of*the classical generalized hexagon H(4). ... In fact, the two-character set is the union*of*two orbits in*PG*(5, 4) under the action*of*L 2 (13). ... Table*3*: Dual codes related to the generalized hexagons*of*order 2. Looking at the tables above, one might try to generalize most*of*these results to arbitrary H(*q*) and/or its dual. ...##
###
Uniform Hyperplanes of Finite Dual Polar Spaces of Rank 3

2001
*
Journal of combinatorial theory. Series A
*

We say that

doi:10.1006/jcta.2000.3136
fatcat:6gnscybpbvg7zbb7yzwjbchxbq
*a*hyperplane H*of*2 is locally singular (respectively, quadrangular or*ovoidal*) if H &*Q*is the perp*of**a*point (resp.*a*subquadrangle or an*ovoid*)*of**Q*for every quad*Q**of*2. ... Let 2 be*a*finite thick dual polar space*of*rank*3*. ... ACKNOWLEDGMENT The authors thank Ernest Shult for having pointed out*a*mistake in an earlier version*of*this paper and for having provided them with Proposition 2.8 and its proof. ...##
###
Classification of 8-dimensional rank two commutative semifields
[article]

2016
*
arXiv
*
pre-print

This is done using computational methods utilizing the connection to linear sets in

arXiv:1606.06151v2
fatcat:pdlefywj25cg5lxnnpc3metlbe
*PG*(2,*q*^4). ... The implications*of*these results are detailed for other geometric structures such as semifield flocks,*ovoids**of*parabolic quadrics, and eggs. ... An*ovoid*O in*Q*(4, s) is*a*translation*ovoid*if there is*a*point p ∈ O and*a*group G*of**collineations**of**Q*(4, s)*stabilizing*O, fixing p, and acting regularly on the points*of*O \{p}. ...##
###
An unusual generalized quadrangle of order sixteen

1978
*
Journal of combinatorial theory. Series A
*

The full

doi:10.1016/0097-3165(78)90044-4
fatcat:twr7g3vxjzdyzmhqkf2hlhistq
*collineation*group*of**3*is determined. ... The generalized quadrangle 9 associated*by*the method*of*Tits with the nonconical*ovoid*discovered*by*M. Hall, Jr., in*pG*(2, 16) is shown to contain no subquadrangle*of*order 4. ... In Section*3*the full*collineation*group*of*9' is determined, with*a**by*-product being that the group*of**collineations**of**PG*(2, 16) leaving L&r' fixed setwise is exactly the subgroup found*by*Hall in ...##
###
Finite elation Laguerre planes admitting a two-transitive group on their set of generators

2013
*
Innovations in Incidence Geometry Algebraic Topological and Combinatorial
*

Equivalently, each translation generalized quadrangle

doi:10.2140/iig.2013.13.207
fatcat:vtzvbqjitrbcpbdmzljes7j22a
*of*order*q*with*a*group*of*automorphisms acting two-transitively on the set*of*lines through the base point is classical. ... We investigate finite elation Laguerre planes admitting*a*group*of*automorphisms that is two-transitive on the set*of*generators. ... The present investigation has been conducted during*a*stay*of*the second author as*a*Visiting Erskine Fellow at the University*of*Canterbury, Christchurch, New Zealand. ...##
###
Page 4584 of Mathematical Reviews Vol. , Issue 94h
[page]

1994
*
Mathematical Reviews
*

De Clerck, Flocks and
51 GEOMETRY
4584
partial flocks

*of*the quadratic cone in*PG*(*3*,*q*) (379-393); J. van Bon, Some extended generalized hexagons (395-403); F. ... For every vertical line g*of*N the mapping 6,:N — N defined*by*6,(P) = (P,N 8): (PN g), is*a**collineation**of*N, which is called*a*Bol reflection. ...##
###
On the existence of O'Nan configurations in Buekenhout unitals in PG(2,q^2)
[article]

2018
*
arXiv
*
pre-print

In this paper, we establish the existence

arXiv:1810.00705v2
fatcat:ztdfjj76ane5rnxunl4i3wn4ea
*of*O'Nan configurations in all nonclassical*ovoidal*Buekenhout-Metz unitals in*PG*(2,*q*^2). ... Acknowledgement This work was supported*by*National Natural Science Foundation*of*China under Grant No. 11771392. Reference ... The odd characteristic case Consider the following involutionary*central**collineation**of*PGL(*3*,*q*2 ): σ : (x, y, z) → (−x, y, z). ...##
###
Contents

2003
*
European journal of combinatorics (Print)
*

-O., On the Eulerian polynomials

doi:10.1016/s0195-6698(03)00149-5
fatcat:4nmkuxvgbzgr7nicrcrobdwvoa
*of*type D . . . . . . 391 BROWN, M.R.,*Ovoids**of**PG*(*3*,*q*)*stabilized**by**a**central**collineation*. 409 MERRIS, R., Split graphs . . . . . . . . . . . . . 413 BOBEN, M. and ... -*Q*. and WANG, K., s-Regular cyclic coverings*of*the threedimensional hypercube*Q**3*. . . . . . . . . . . . 719 DELANOTE, M., Point derivation*of*partial geometries . . . . . . 733 WILDBERGER, N.J., Minuscule ...##
###
Index

2003
*
European journal of combinatorics (Print)
*

tree-like equalities . . . . . . .
557
BROWN, M.R.,

doi:10.1016/s0195-6698(03)00150-1
fatcat:q6zy3wos7vdkzfy55qdwge5nl4
*Ovoids**of**PG*(*3*,*q*)*stabilized**by**a**central**collineation*. 409 BROWN, M.R., DE BEULE, J. and STORME, L., Maximal partial spreads*of*T 2 (O) and ... -*Q*. and WANG, K., s-Regular cyclic coverings*of*the threedimensional hypercube*Q**3*. . . . . . . . . . . 719 FIORINI, S.,*A*combinatorial study*of*partial order polytopes . . . . 149 FRANCHI, C. and VSEMIRNOV ...
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