André plane explained
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
be a finite
field, and let
be a degree
extension field of
. Let
be the group of field automorphisms of
over
, and let
be an arbitrary mapping from
to
such that
. Finally, let
be the
norm function from
to
.
Define a quasifield
with the same elements and addition as K, but with multiplication defined via
, where
denotes the normal field multiplication in
. Using this quasifield to construct a plane yields an André plane.
Properties
- André planes exist for all proper prime powers
with
prime and
a positive integer greater than one.
- Non-Desarguesian André planes exist for all proper prime powers except for
where
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.[1]
- 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.[2]
- There is a single non-Desarguesian André plane of order 27.[3]
Enumeration of Andrè planes specifically has been performed for other small orders:[4]
| Order | Number ofnon-Desarguesian Andrè planes
|
|---|
| 9 | 1 |
| 16 | 1 |
| 25 | 3 |
| 27 | 1 |
| 49 | 7 |
| 64 | 6 (four 2-d, two 3-d) |
| 81 | 14 (13 2-d, one 4-d) |
| 121 | 43 |
| 125 | 6 | |
Notes and References
- Web site: Projective Planes of Order 16. 2020-11-08. ericmoorhouse.org.
- Dover. Jeremy M.. 2019-02-27. A genealogy of the translation planes of order 25. math.CO. 1902.07838.
- Web site: Projective Planes of Order 27. 2020-11-08. ericmoorhouse.org.
- Dover. Jeremy M.. 2021-05-16. Computational Enumeration of Andrè Planes. math.CO. 2105.07439.
