Unitals in the Hall plane

1997 ◽  
Vol 58 (1-2) ◽  
pp. 26-42 ◽  
Author(s):  
S. G. Barwick
Keyword(s):  
Hall Plane

Collineation Groups of Generalized André Planes

1969 ◽  
Vol 21 ◽  
pp. 358-369 ◽  
Author(s):  
David A. Foulser
Keyword(s):  
Main Question ◽  
Translation Planes ◽  
Hall Plane ◽  
Collineation Groups

In a previous paper (5), I constructed a class of translation planes, called generalized André planes or λ-planes, and discussed the associated autotopism collineation groups. The main question unanswered in (5) is whether or not there exists a collineation η of a λ-plane Π which moves the two axes of Π but does not interchange them.The answer to this question is “no”, except if Π is a Hall plane (or possibly if the order n of Π is 34) (Corollary 2.8). This result makes it possible to determine the isomorphisms between λ-planes. More specifically, let Π and Π′ be two λ-planes of order n coordinatized by λ-systems Qand Q′, respectively. Then, except possibly if n = 34, Π and Π′ are isomorphic if and only if Q and Q′ are isotopic or anti-isotopic (Corollary 2.13). In particular, Π is an André plane if and only if Q is an André system (Corollary 2.14).

scholarly journals On Conics that are Ovals in a Hall Plane

1993 ◽  
Vol 14 (6) ◽  
pp. 521-528 ◽  
Author(s):  
D.G. Glynn ◽  
G.F. Steinke
Keyword(s):  
Hall Plane

scholarly journals A characterization of generalized Hall planes

1972 ◽  
Vol 6 (1) ◽  
pp. 61-67 ◽  
Author(s):  
N.L. Johnson
Keyword(s):  
Translation Plane ◽  
Hall Plane ◽  
Odd Order ◽  
Class 1 ◽  
Even Order ◽  
Semifield Planes ◽  
Dickson Semifield

We prove that a translation plane π of odd order is a generalized Hall plane if and only if π is derived from a translation plane of semi-translation class 1–3a. Also, a derivable translation plane of even order and class 1–3a derives a generalized Hall plane. We also show that the generalized Hall planes of Kirkpatrick form a subclass of the class of planes derived from the Dickson semifield planes.

Geometric Axioms for the Hall Plane

1973 ◽  
Vol s2-6 (2) ◽  
pp. 351-357
Author(s):  
T. G. Room
Keyword(s):  
Hall Plane

scholarly journals On elations in semi-transitive planes

1982 ◽  
Vol 5 (1) ◽  
pp. 159-164
Author(s):  
N. L. Johnson
Keyword(s):  
Translation Plane ◽  
Hall Plane ◽  
Even Order ◽  
Affine Elation

Letπbe a semi-transitive translation plane of even order with reference to the subplaneπ0. Ifπadmits an affine elation fixingπ0for each axis inπ0and the order ofπ0is not2or8, thenπis a Hall plane.

On points and lines of the left Hall plane of order 9

2015 ◽  
Vol 9 ◽  
pp. 537-542
Author(s):  
Z. Akca ◽  
A. Bayar ◽  
M. Tas
Keyword(s):  
Hall Plane

scholarly journals The existence of Fano subplanes in generalized Hall planes

1973 ◽  
Vol 16 (2) ◽  
pp. 234-238 ◽  
Author(s):  
A. J. Rahilly
Keyword(s):  
Sufficient Condition ◽  
Hall Plane ◽  
Desarguesian Planes ◽  
Old Order ◽  
Odd Order

One of the best known classes of non-Desarguesian planes is the calss of Hall planes (see Hall [2]). In [6] Hanna Neumann showed that the finite Hall planes of old order possess subplanes of order two (i.e., Fano subplanes)1. Kirkpatrick [5] has considered a type of plane which is generalization of the Hall planes and which he calls generalized Hall planes. In this paper we will give a sufficient condition that a finite generalized Hall plane possesses Fano subplanes. Some examples of odd order planes to which the condition applies shall be exhibited.

On Translation Planes which Admit Solvable Autotopism Groups Having a Large Slope Orbit

1984 ◽  
Vol 36 (5) ◽  
pp. 769-782 ◽  
Author(s):  
Vikram Jha
Keyword(s):  
Translation Plane ◽  
Collineation Group ◽  
Affine Plane ◽  
Translation Planes ◽  
Hall Plane ◽  
Autotopism Group ◽  
Affine Translation ◽  
Planar Group ◽  
Affine Translation Plane

Our main object is to prove the following result.THEOREM C. Let A be an affine translation plane of order qr ≧ q2 suchthatl∞, the line at infinity, coincides with the translation axis of A. Suppose G is a solvable autotopism group of A that leaves invariant a set Δ of q + 1 slopes and acts transitively on l∞ \ Δ.Then the order of A is q2.An autotopism group of any affine plane A is a collineation group G that fixes at least two of the affine lines of A; if in fact the fixed elements of G form a subplane of A we call G a planar group. When A in the theorem is a Hall plane [4, p. 187], or a generalized Hall plane ([13]), G can be chosen to be a planar group.

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