Collineation groups preserving a unital in a projective plane of even order

1989 ◽  
Vol 31 (3) ◽  
Author(s):  
Mauro Biliotti ◽  
Gabor Korchmaros
Keyword(s):  
Projective Plane ◽  
Collineation Groups ◽  
Even Order

scholarly journals Maximal arcs in projective planes of order 16 and related designs

2019 ◽  
Vol 19 (3) ◽  
pp. 345-351 ◽  
Author(s):  
Mustafa Gezek ◽  
Vladimir D. Tonchev ◽  
Tim Wagner
Keyword(s):  
Projective Plane ◽  
Projective Planes ◽  
Maximal Arcs ◽  
Even Order

Abstract The resolutions and maximal sets of compatible resolutions of all 2-(120,8,1) designs arising from maximal (120,8)-arcs, and the 2-(52,4,1) designs arising from previously known maximal (52,4)-arcs, as well as some newly discovered maximal (52,4)-arcs in the known projective planes of order 16, are computed. It is shown that each 2-(120,8,1) design associated with a maximal (120,8)-arc is embeddable in a unique way in a projective plane of order 16. This result suggests a possible strengthening of the Bose–Shrikhande theorem about the embeddability of the complement of a hyperoval in a projective plane of even order. The computations of the maximal sets of compatible resolutions of the 2-(52,4,1) designs associated with maximal (52,4)-arcs show that five of the known projective planes of order 16 contain maximal arcs whose associated designs are embeddable in two nonisomorphic planes of order 16.

scholarly journals Collineation groups preserving an oval in a projective place of odd order

1990 ◽  
Vol 48 (1) ◽  
pp. 156-170 ◽  
Author(s):  
Mauro Biliotti ◽  
Gabor Korchmaros
Keyword(s):  
Projective Plane ◽  
Collineation Group ◽  
Finite Projective Plane ◽  
Odd Order ◽  
Collineation Groups

AbstractIn this paper we investigate the structure of a collineation group G of a finite projective plane Π of odd order, assuming that G leaves invariant an oval Ω of Π. We show that if G is nonabelian simple, then G ≅ PSL(2, q) for q odd. Several results about the structre and the action of G are also obtained under the assumptions that n ≡ 1 (4) and G is transitive on the points of Ω.

scholarly journals Distance-Restricted Matching Extension in Triangulations of the Torus and the Klein Bottle

10.37236/2952 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Robert E.L. Aldred ◽  
Jun Fujisawa
Keyword(s):  
Projective Plane ◽  
Perfect Matching ◽  
Klein Bottle ◽  
Even Order

A graph $G$ with at least $2m+2$ edges is said to be distance $d$ $m$-extendable if for any matching $M$ in $G$ with $m$ edges in which the edges lie pair-wise distance at least $d$, there exists a perfect matching in $G$ containing $M$. In a previous paper, Aldred and Plummer proved that every $5$-connected triangulation of the plane or the projective plane of even order is distance $5$ $m$-extendable for any $m$. In this paper we prove that the same conclusion holds for every triangulation of the torus or the Klein bottle.

On Baer Involutions of Finite Projective Planes

1970 ◽  
Vol 22 (4) ◽  
pp. 878-880 ◽  
Author(s):  
Judita Cofman
Keyword(s):  
Projective Plane ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Desarguesian Projective Plane ◽  
Finite Projective Planes ◽  
Special Cases ◽  
Collineation Groups ◽  
Baer Involution ◽  
Finite Desarguesian Projective Plane

1. An involution of a projective plane π is a collineation X of π such that λ2 = 1. Involutions play an important röle in the theory of finite projective planes. According to Baer [2], an involution λ of a finite projective plane of order n is either a perspectivity, or it fixes a subplane of π of order in the last case, λ is called a Baer involution.While there are many facts known about collineation groups of finite projective planes containing perspectivities (see for instance [4; 5]), the investigation of Baer involutions seems rather difficult. The few results obtained about planes admitting Baer involutions are restricted only to special cases. Our aim in the present paper is to investigate finite projective planes admitting a large number of Baer involutions. It is known (see for instance [3, p. 401]) that in a finite Desarguesian projective plane of square order, the vertices of every quadrangle are fixed by exactly one Baer involution.

scholarly journals Cubic Extensions of Flag-Transitive Planes, I. Even Order

2001 ◽  
Vol 25 (8) ◽  
pp. 533-547 ◽  
Author(s):  
Yutaka Hiramine ◽  
Vikram Jha ◽  
Norman L. Johnson
Keyword(s):  
Translation Planes ◽  
Collineation Groups ◽  
Even Order

The collineation groups of even order translation planes which are cubic extensions of flag-transitive planes are determined.

Collineation Groups Containing Perspectivities

1967 ◽  
Vol 19 ◽  
pp. 924-937 ◽  
Author(s):  
Peter Dembowski
Keyword(s):  
Projective Plane ◽  
Finite Order ◽  
Translation Plane ◽  
Collineation Groups

Let P be a projective plane of finite order n and Γ a group of collineations of P. Gleason (6) and Wagner (10) have shown that if every point of P is the centre, and every line the axis, of a non-trivial perspectivity in Γ, then Γ contains a subgroup of order n2 which consists entirely of elations. It then follows that either P or its dual is a translation plane with respect to at least one line; in fact if Γ has no fixed elements, then P is desarguesian and Γ contains all elations of P. It was shown by Piper (7) and Cofman (4) that the hypotheses of Gleason and Wagner can be relaxed in certain cases, while the same conclusions hold.

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