Translation Planes Of Order Research Articles

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

A translation plane of order 72 is constructed whose left nucleus is a cyclic group of order 6. Generally the translation complement of a translation plane is quite large when compared to the order of the left nucleus. The plane discussed in this paper has the distinguished feature that its translation complement modulo the scalar collineations group is very small. In fact it is a Dihedral group of order 12 and is the smallest of all the known examples. The translation complement divides the set of ideal points into 9 orbits of lengths 1,1, 6, 6, 6, 6, 6, 6, and 12.

Near nearfield planes

The translation planes of order k that admit at least two affine homology groups of order (k−1)/2 are classified. If k≠72 or 232, the plane is generalized Andre and there is a Dickson pair {q, n} such that qn=k and a cyclic homology subgroup of order (qn−1)/2n.

Translation planes of order 25 admitting a quaternion group of homologies with affine axis

Translation planes of order 25 admitting a quaternion group of homologies with affine axis

A family of translation planes of order q 2m+1 with two orbits of length 2 and q 2m+1?1 on l ?

A class of translation planes of order q 2m+1, where q is an odd prime power and m≥1, is constructed. If m=1, then this class is contained in the class of order q 3 constructed by Hiramine [5]. These planes of order q 2m+1 are of dimension 2m+1 over their kernels. If q 2m+1≠33, then the linear translation complements of these planes have two orbits of length 2 and q 2m+1−1 on l ∞ and this class contains many planes which are not generalized Andre planes. If q 2m+1= 33, then each plane of this class is isomorphic to the Hering plane of order 27.

Some translation planes of order $\text{11}^{2}$ which admit SL(2,9)

Some translation planes of order $\text{11}^{2}$ which admit SL(2,9)

A non-symmetric translation plane of order 172

A non-symmetric translation plane of order 172

A characterization of Hering's plane of order 27

Hering's translation plane of order 27 has been characterized by its order and the fact that it admits SL(2, 13) in its translation complement (see [1]). We show that, aside from the Desarguian plane and a Generalized, Andre plane, it is the only plane of order 27 which admits a subgroup of SL(2, 13) of order 13×12.

A structure theory for two-dimensional translation planes of order q 2 that admit collineation groups of order q 2

This paper is devoted to the study of translation planes of order q2 and kernel GF(q) that admit a collineation group of order q2 in the linear translation complement. We give a representation of this group by a suitable set of matrices depending on some functions over GF(q). Using this representation we obtain several results concerning the existence and the collineation group of the plane.

Translationsebenen der ordnung 49 mit scherungen

It is shown that there exists, up to isomorphism, exactly one non-Desarguesian translation plane of order 49 which admits non-trivial shears. Furthermore, the collineation groups of this plane and all its derived planes are described.

Regular parallelisms from translation planes

Let π be a translation plane of order q 4 admitting a collineation group G isomorphic to SL(2, q), p r = q, in the translation complement. It is shown that if the p-elements of G are elations and L is an elation axis then the kernel of π is GF( q) and G acts 1 2 - transitive on l ∞− N∩l ∞ (where N denotes the elation net) if and only if there is a regular parallelism of L (considering L as PG(3, q)). With no assumption on the p-elements, it is shown that the existence SL(2, q)× z 1+ q+ q 2 as a collineation group implies that the p-elements are elations, the kernel is GF( q) and any elation axis L admits a regular parallelism as PG(3, q).

A Note on Net Replacement in Transposed Spreads

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).

A note on finite semifield planes that admit affine homologies

Let π be a translation plane of order n that admits an abelian group of order n in its translation complement. If π admits an affine homology then it is a semifield plane.

A translation plane of order 49 with non-solvable collineation group

A translation plane of order 49 with non-solvable collineation group

On a generalization of Kantor's likeable planes

Generalizing an idea of Kantor [7], Johnson and Wilke [5] introduced ‘elusive’ sets of functions over GF(q) to represent translation planes of order q2 that admit a collineation group of order q2 in the linear translation complement and whose kern contains GF(q). In this paper we determine explicitly all elusive sets for q even. We obtain another translation plane of order 82.

Some translation planes of order p 2 and of extra-special type

Some translation planes of order p 2 and of extra-special type

Some translation planes of order 72 which admit SL2(9)

Some translation planes of order 72 which admit SL2(9)

On translation planes of order $q\sp 2$ which admit an autotopism group having an orbit of length $q\sp 2-q$

On translation planes of order $q\sp 2$ which admit an autotopism group having an orbit of length $q\sp 2-q$

Some unusual translation planes of order 64

Some unusual translation planes of order 64

The classification of the translation planes of order 16, II

The translation planes of order 16 are completely classified. The exceptional isomorphism A8≃GL(4, 2) gives a crucial computational approach to this problem

A translation plane of order 25

In this paper a new translation plane of order 25 is constructed. It has a collineation group acting on the line at infinity as a permutation group Z of order 48 with the properties: (i) Z contains a normal subgroup 1/2M of order 3 such that Z/1/2M is the direct product of an involution with a dihedral group of order 8. (ii) The orbits of Z have lengths 2, 12, 12.