A complete classification of the flag-transitive affine planes of order 125

The flag-transitive affine planes of order 125 are completely classified. There are five such planes. © 1997 John Wiley & Sons, Inc.     

Pathological projective planes: Associate affine planes

In [7] the author showed the existence of projective plane pathological with respect to the collineation groups of its sub and quotient planes. Similar pathologies are obtainable with respect to collineation groups of associated affine planes. (i.e. the affine planes obtained by distinguishing a line as the line at infinity) as expressable in the following theorem.     

Extended dual affine planes

We study a class of diagram geometries, achieve a characterization of extended dual affine planes, and embed extended dual affine planes in extended projective planes. The geometries studied are rank 3 diagram geometries such that the residue of a point is a dual net, and the residue of a plane is linear; the dual of such a geometry has partitions on lines and planes which are reminiscent of parallelism of lines and planes of an affine 3-space. Examples of these geometries (some in dual form) include extended dual affine planes, Laguerre planes, 3-nets, and orthogonal arrays of strength 3. Theorem: Any such finite geometry satisfying Buekenhout's intersection property, and such that any two points are coplanar, is an extended dual affine plane (and has order 2, 4, or 10). Theorem: This geometry may be embedded in an extended projective plane of the same order.This research was partially supported by NSF Grant MCS-8102361.     

On point-transitive affine planes

Finite affine planes are constructed admitting nonabelian sharply point-transitive collineation groups. These planes are of two sorts: dual translation planes, and planes of type II.1 derived from them. This research was supported in part by National Science Foundation Grant MCS-7903130.     

Ordered affine Hjelmslev planes

Preorderings and orderings of affine Hjelmslev planes were defined in [3]. The ordering of an AH-plane is shown to induce an ordering on each coordinate biternary ring which is compatible with the multiplication of the associated algebra. Even stronger projective order relations, on both the planes and biternary rings, are introduced and are shown to be also compatible with the ternary operations.The author gratefully acknowledges the support of the Natural Sciences and Engineering Research Council of Canada, and the invaluable assistance of Dr. N. D. Lane, McMaster University.     

Preordered affine Hjelmslev planes

An affine Hjelmslev plane (AH-plane)H is preordered if it is endowed with a betweenness relation which is preserved by bijective parallel projections. Preordered biternary rings are defined and are used to construct preordered AH-planes. Conversely, a preordered AH-plane induces a preordering on any of its biternary rings. The concepts used in this paper are compared with those in a recent independent study of preordered affine Klingenberg planes by F. Machala. Unlike the Klingenberg case, preordered AH-planes always possess convex neighbour classes and unlike ordered ordinary affine planes, preordered AH-planes are never archimedian.The authors gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada.     

Near affine Hjelmslev planes

A study is made of two generalizations of affine Hjelmslev planes in which the parallel axiom is not required to hold. Integer invariants are obtained for the finite planes in these new classes. Formulas are derived which enable one to compute the cardinalities of certain subsets of points and lines in terms of the invariants, and results are obtained on the nonexistence of planes with certain sets of invariants.     

Subplane covered affine planes

Projective or affine planes that are covered by subplanes in a prescribed manner are studied and classified. This provides a generalization of the concept of regular spread.This research begun while the second author was a C.N.R. visiting professor in Italy during May–June 1994.     

Derived semifield planes of odd order admitting a dihedral group of affine homologies

Derived semifield planes of odd order admitting non-trivial involutorial affine homologies with more than one axis, are examined in detail, under the assumption that the group generated in the translation complement is dihedral. The whole structure of the semifields S coordinatizing such planes is determined. The class of the semifields S of dimension 4 over their centres is characterized.Dedicated to A. Barlotti on the occasion of his 65. birthday.Research partially supported by G.N.S.A.G.A. (C.N.R.)     

Transitive affine planes admitting elations

Affine planes which admit a point transitive collineation group and at least one affine elation are considered. Such a plane is shown to be (A,?_∞)-transitive for some point A on ?_t8 and to be a translation plane if at least two distinct elation centers exist. If the plane has at least (order)^1/2+1 distinct elation centers and the group generated by the elations is nonsolvable then the plane is either Desarguesian or Lüneburg-Tits.     

Point homogeneous flat affine planes

All flat projective planes (P,) whose automorphism group contains a 2-dimensional, connected closed subgroup fixing at most one line 1, are classified, except the following 2 classes: (1) – L₂, and not transitive on P1, and (2) S₁xR.Research supported in part by National Research Council of Canada (NRC)Herrn Professor Dr. Werner Burau zum 70. Geburtstag     

Area functions on affine planes

Oriented area functions are functions defined on the set of ordered triangles of an affine plane which are antisymmetric under odd permutations of the vertices and which behave additively when triangles are cut into two. We compare several elementary properties which such an area function may have (roughly speaking shear invariance, equality of area of the two triangles obtained by cutting a parallelogram along a diagonal, and equality of area of the two triangles obtained by cutting a triangle along a median). It turns out purely by arguments of elementary affine geometry (if cleverly arranged) that these properties are grosso modo equivalent, although one has to be careful about “pathological” situations. Furthermore, all oriented area functions satisfying these properties are explicitly determined. Finally they are compared with so-called geometric valuations.     

Characterization of symmetric planes in dimension at most 4

A stable plane is a topological geometry with the properties that (i) any two points are joined by a unique line, and (ii) the operations of join and intersection are continuous and have open domains of definition. A stable plane is called symmetric if its point space is a (differentiable) symmetric space whose symmetries are automorphisms of the plane.Among locally compact stable planes of positive (topological) dimension ≦ 4, we determine those which admit a reflection at each point (i.e., an involutory automorphism fixing this point line-wise), and we list the possible groups containing reflections at all points. Together with an additional, purely geometric condition, this yields a characterization of symmetric planes and, indirectly, of the plane geometries defined by real and complex hermitian forms. No differentiability hypotheses and no algebraic axioms ruling the reflections are needed.