A non-symmetric translation plane of order 172

1990 ◽  
Vol 37 (1-2) ◽  
pp. 77-83 ◽  
Author(s):  
Chris Charnes
Keyword(s):  
Translation Plane

On a conjecture of Hughes

1967 ◽  
Vol 63 (3) ◽  
pp. 647-652 ◽  
Author(s):  
Judita Cofman
Keyword(s):  
Projective Plane ◽  
Permutation Group ◽  
Translation Plane ◽  
Collineation Group ◽  
Affine Plane ◽  
Finite Projective Plane ◽  
Transitive Permutation Group ◽  
Condition C

D. R. Hughes stated the following conjecture: If π is a finite projective plane satisfying the condition: (C)π contains a collineation group δ inducing a doubly transitive permutation group δ* on the points of a line g, fixed under δ, then the corresponding affine plane πg is a translation plane.

scholarly journals A translation plane of order 112

1983 ◽  
Vol 35 (3) ◽  
pp. 289-300 ◽  
Author(s):  
Mauro Capursi
Keyword(s):  
Translation Plane

Extension of collineations defined on subsets of a translation plane

1971 ◽  
Vol 1 (1) ◽  
pp. 1-17 ◽  
Author(s):  
F. Radó
Keyword(s):  
Translation Plane

scholarly journals On collineation groups of translation planes of orderq4

1986 ◽  
Vol 9 (3) ◽  
pp. 617-620
Author(s):  
V. Jha ◽  
N. L. Johnson
Keyword(s):  
Translation Plane ◽  
Translation Planes ◽  
Nonsolvable Group ◽  
Translation Complement ◽  
Affine Translation ◽  
Collineation Groups ◽  
Affine Translation Plane

LetPbe an affine translation plane of orderq4admitting a nonsolvable groupGin its translation complement. IfGfixes more thanq+1slopes, the structure ofGis determined. In particular, ifGis simple thenqis even andG=L2(2s)for some integersat least2.

A Note on Net Replacement in Transposed Spreads

1985 ◽  
Vol 28 (4) ◽  
pp. 469-471 ◽  
Author(s):  
N. L. Johnson
Keyword(s):  
Translation Plane ◽  
Translation Planes ◽  
Net Replacement

AbstractLet P be a translation plane containing a net N which is replaceable by . Let P' denote the transposed plane. We note that N' is replaceable by ()'. This result shows how to relate the various constructions of the two translation planes of order 16 that admit PSL(2, 7).

Moulton Planes

1961 ◽  
Vol 13 ◽  
pp. 427-436 ◽  
Author(s):  
William A. Pierce
Keyword(s):  
Translation Plane ◽  
Affine Plane ◽  
Negative Slope ◽  
Arbitrary Field ◽  
Desarguesian Plane ◽  
Real Numbers ◽  
Classical Plane ◽  
Real Affine ◽  
Desargues Theorem ◽  
Ordered Pairs

In 1902, F. R. Moulton (12) gave an early example of a non-Desarguesian plane. Its ‘points” are ordered pairs (x, y) of real numbers. Its “lines” coincide with lines of the real affine plane except that lines of negative slope are “bent” on the x-axis, line {y = b + mx}, for negative m, being replaced by {y = b + mx if y ≤ 0, y = [m/2]. [x + (b/m)] if y > 0}. A certain Desarguesian configuration in the classical plane is shifted just enough to vitiate Desargues’ Theorem for Moulton's geometry. The plane is neither a translation plane (“Veblen-Wedderburn” in the sense of Hall (7), p. 364) nor even the dual of one (Veblen and Wedderburn (17). It is natural to ask if the same construction is feasible when real numbers are replaced by elements from an arbitrary field.

scholarly journals A translation plane of order 25 and its full collineation group

1978 ◽  
Vol 19 (3) ◽  
pp. 351-362 ◽  
Author(s):  
M.L. Narayana Rao ◽  
K. Kuppuswamy Rao
Keyword(s):  
Translation Plane ◽  
Collineation Group ◽  
Translation Planes ◽  
Single Orbit ◽  
Full Collineation Group ◽  
Ideal Points ◽  
The Ideal

Ostrom proposed classifications of translation planes on the basis of the action of the collineation group of the plane on the ideal points. There are examples of translation planes in which ideal points form a single orbit (flag transitive planes) and also several orbits (Hall, André, Foulser, and so forth, planes). In this paper the authors have constructed a translation plane in which the ideal points are divided into two orbits of lengths 18 and 8 respectively. A few collineatlons are computed together with their actions. The group of collineations G1 which is transitive on the two sets of 18 and 8 lines separately is calculated. All the collineations that fix L0 are also calculated and they form a group of. If G2 is the group of translations then the full collineation group is shown to be 〈G1, G2, G3〉.

scholarly journals An upper bound for the p-rank of a translation plane

1991 ◽  
Vol 56 (2) ◽  
pp. 297-302 ◽  
Author(s):  
Jennifer D Key ◽  
Kirsten Mackenzie
Keyword(s):  
Upper Bound ◽  
Translation Plane

Subplanes of the Hughes plane of order 9

1968 ◽  
Vol 64 (3) ◽  
pp. 589-598 ◽  
Author(s):  
R. H. F. Denniston
Keyword(s):  
Translation Plane ◽  
Projective Planes ◽  
Point Of View ◽  
Finite Plane ◽  
Field Of Study ◽  
Finite Projective Planes ◽  
Hughes Plane

The literature of finite projective planes consists largely of general investigations, taking in as many as possible of these systems at once. However, the geometry in a specific finite plane may well be an amusing, and not entirely trivial, field of study on its own. Some papers (5, 9, 13) have in fact appeared on the geometry of the translation plane of order 9: but the Hughes plane of the same order has comparatively been neglected. The object of the present paper is to make a beginning with the study of this plane from a synthetic point of view.

A translation plane of order 72 with a very small translation complement

1993 ◽  
Vol 9 (2-4) ◽  
pp. 255-263 ◽  
Author(s):  
M. L. Narayana Rao ◽  
K. Kuppu Swamy Rao ◽  
Vinod Joshi
Keyword(s):  
Translation Plane ◽  
Translation Complement
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