scholarly journals Line-rank 3 affine planes

1982 ◽  
Vol 25 (3) ◽  
pp. 397-403
Author(s):  
Michael J. Kallaher ◽  
Graham Kelly
Keyword(s):  
Permutation Group ◽  
Collineation Group ◽  
Classical Result ◽  
Translation Planes ◽  
Affine Planes ◽  
Finite Affine ◽  
Dimension 2

We consider finite affine planes having a collineation group acting as a rank 3 permutation group on the affine lines. By a classical result of A. Wagner, such affine planes are translation planes. We show that if, in addition, the plane has odd dimension or dimension 2 over its kernel, then the plane is Desarguesian.

scholarly journals Semi-field-like affine planes

1977 ◽  
Vol 18 (2) ◽  
pp. 113-123 ◽  
Author(s):  
Michael J. Kallaher
Keyword(s):  
Translation Plane ◽  
Collineation Group ◽  
The Other ◽  
Translation Planes ◽  
Affine Planes ◽  
New Class

Walker [11] describes a new class of translation planes W(q) of order q2, q ≡ 5(mod 6), with kernel GF(q). A plane in this class has several interesting properties, but we shall be only interested in the following ones possessed by its collineation group G: (i) G is transitive on the affine points of W(q), and (ii) G fixes a point the line at infinity of W(q), and is transitive on the other points of l∞ The smallest member of this class, W(5), also satisfies: (iii) for an affine point the subgroup G is transitive on the affine points ≠ of the line Note also that since W(5) is a translation plane, replacing G with G in (ii) we get a fourth property, call it (iv), satisfied by G. (See Lemma 2.)

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.

Order in Finite Affine Planes

1966 ◽  
Vol 9 (4) ◽  
pp. 407-411
Author(s):  
Mary Beattie ◽  
J.E. Marsden ◽  
R. W. Sharpe
Keyword(s):  
Affine Planes ◽  
Finite Affine

An ordered plane [1, p. 177 ff.] is a set of undefined entities called points, together with an undefined relation of intermediacy, satisfying the following axioms: (The symbol [ABC] means ″B is between A and C″.)

scholarly journals On Finite Affine Planes with a 2-Transitive Orbit on l∞

1993 ◽  
Vol 162 (2) ◽  
pp. 392-409 ◽  
Author(s):  
Y. Hiramine
Keyword(s):  
Affine Planes ◽  
Finite Affine

scholarly journals BRUCK NETS AND PARTIAL SHERK PLANES

2017 ◽  
Vol 104 (1) ◽  
pp. 1-12
Author(s):  
JOHN BAMBERG ◽  
JOANNA B. FAWCETT ◽  
JESSE LANSDOWN
Keyword(s):  
Affine Plane ◽  
Finite Geometries ◽  
Affine Planes ◽  
Incidence Structures ◽  
Finite Affine ◽  
Odd Order ◽  
Finite Incidence ◽  
Desarguesian Affine Plane

In Bachmann [Aufbau der Geometrie aus dem Spiegelungsbegriff, Die Grundlehren der mathematischen Wissenschaften, Bd. XCVI (Springer, Berlin–Göttingen–Heidelberg, 1959)], it was shown that a finite metric plane is a Desarguesian affine plane of odd order equipped with a perpendicularity relation on lines and that the converse is also true. Sherk [‘Finite incidence structures with orthogonality’, Canad. J. Math.19 (1967), 1078–1083] generalised this result to characterise the finite affine planes of odd order by removing the ‘three reflections axioms’ from a metric plane. We show that one can obtain a larger class of natural finite geometries, the so-called Bruck nets of even degree, by weakening Sherk’s axioms to allow noncollinear points.

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