André plane

In mathematics, André planes are a class of finite translation planes found by André.^[1] The Desarguesian plane and the Hall planes are examples of André planes; the two-dimensional regular nearfield planes are also André planes.

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

Define a quasifield ${\displaystyle Q}$ with the same elements and addition as K, but with multiplication defined via ${\displaystyle a\circ b=a^{\beta (N(b))}\cdot b}$, where ${\displaystyle \cdot }$ denotes the normal field multiplication in ${\displaystyle K}$. Using this quasifield to construct a plane yields an André plane.^[2]

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

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.^[3]
• There are three non-Desarguesian André planes of order 25.^[4] These are the Hall plane, the regular nearfield plane, and a third plane not constructible by other techniques.^[5]
• There is a single non-Desarguesian André plane of order 27.^[6]

Enumeration of Andrè planes specifically has been performed for other small orders:^[7]