André plane

In mathematics, André planes are a class of finite translation planes found by André. The Desarguesian plane and the Hall planes are examples of André planes; the two-dimensional regular nearfield planes are also André planes.

Construction
Let $$F = GF(q)$$ be a finite field, and let $$K = GF(q^n)$$ be a degree $$n$$ extension field of $$F$$. Let $$\Gamma$$ be the group of field automorphisms of $$K$$ over $$F$$, and let $$\beta$$ be an arbitrary mapping from $$F$$ to $$\Gamma$$ such that $$\beta(1)=1$$. Finally, let $$N$$ be the norm function from $$K$$ to $$F$$.

Define a quasifield $$Q$$ with the same elements and addition as K, but with multiplication defined via $$a \circ b = a^{\beta(N(b))} \cdot b$$, where $$\cdot$$ denotes the normal field multiplication in $$K$$. Using this quasifield to construct a plane yields an André plane.

Properties

 * 1) André planes exist for all proper prime powers $$p^n$$ with $$p$$ prime and $$n$$ a positive integer greater than one.
 * 2) Non-Desarguesian André planes exist for all proper prime powers except for $$2^n$$ where $$n$$ is prime.

Small Examples
For planes of order 25 and below, classification of Andrè planes is a consequence of either theoretical calculations or computer searches which have determined all translation planes of a given order:


 * The smallest non-Desarguesian André plane has order 9, and it is isomorphic to the Hall plane of that order.
 * The translation planes of order 16 have all been classified, and again the only non-Desarguesian André plane is the Hall plane.
 * There are three non-Desarguesian André planes of order 25. These are the Hall plane, the regular nearfield plane, and a third plane not constructible by other techniques.
 * There is a single non-Desarguesian André plane of order 27.

Enumeration of Andrè planes specifically has been performed for other small orders: