The central collineations with center P and axis l form a group.
[2] A line l in a projective plane Π is a translation line if the group of all elations with axis l acts transitively on the points of the affine plane obtained by removing l from the plane Π, Πl (the affine derivative of Π).
[3][4] Every projective plane can be coordinatized by at least one planar ternary ring.
[6] However, some quasifields satisfy additional algebraic properties, and the corresponding planar ternary rings coordinatize translation planes which admit additional symmetries.
(multiplication), one can define a planar ternary ring to create coordinates for a translation plane.
However, it is more typical to create an affine plane directly from the quasifield by defining the points as pairs
[7] Translation planes are related to spreads of odd-dimensional projective spaces by the Bruck-Bose construction.
is an integer and K a division ring, is a partition of the space into pairwise disjoint n-dimensional subspaces.
In the finite case, a spread of PG(2n+1, q) is a set of qn+1 + 1 n-dimensional subspaces, with no two intersecting.
In the finite case, this procedure produces a translation plane of order qn+1.
[9] Any translation plane which is coordinatized by a quasifield that is finite-dimensional over its kernel K (K is necessarily a division ring) can be generated from a spread of PG(2n+1, K) using the Bruck-Bose construction, where (n+1) is the dimension of the quasifield, considered as a module over its kernel.
An instant corollary of this result is that every finite translation plane can be obtained from this construction.
André[10] gave an earlier algebraic representation of (affine) translation planes that is fundamentally the same as Bruck/Bose.
Let V be a 2n-dimensional vector space over a field F. A spread of V is a set S of n-dimensional subspaces of V that partition the non-zero vectors of V. The members of S are called the components of the spread and if Vi and Vj are distinct components then Vi ⊕ Vj = V. Let A be the incidence structure whose points are the vectors of V and whose lines are the cosets of components, that is, sets of the form v + U where v is a vector of V and U is a component of the spread S. Then:[11] Let F = GF(q) = Fq, the finite field of order q and V the 2n-dimensional vector space over F represented as: Let M0, M1, ..., Mqn - 1 be n × n matrices over F with the property that Mi – Mj is nonsingular whenever i ≠ j.
is regular if for any three distinct n-dimensional subspaces of S, all the members of the unique regulus determined by them are contained in S. For any division ring K with more than 2 elements, if a spread S of PG(2n+1, K) is regular, then the translation plane created by that spread via the André/Bruck-Bose construction is a Moufang plane.
A slightly weaker converse holds: if a translation plane is Pappian, then it can be generated via the André/Bruck-Bose construction from a regular spread.
, and the classes of Moufang, Desarguesian and Pappian planes are all identical, so this theorem can be refined to state that a spread S of PG(2n+1, q) is regular if and only if the translation plane created by that spread via the André/Bruck-Bose construction is Desarguesian.
, all spreads of PG(2n+1, 2) are trivially regular, since a regulus only contains three elements.
[16] Finite translation planes must have prime power order.
The following table details the current state of knowledge: Translation Planes