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.

Construction

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]

Properties

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.

Small Examples

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 André plane of order 27.[6]

References

1. André, Johannes (1954), "Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe", Mathematische Zeitschrift, 60: 156–186, doi:10.1007/BF01187370, ISSN 0025-5874, MR 0063056, S2CID 123661471
2. Weibel, Charles (2007), "Survey of Non-Desarguesian Planes", Notices of the AMS, 54 (10): 1294–1303
3. "Projective Planes of Order 16". ericmoorhouse.org. Retrieved 2020-11-08.
4. Chen, G. (1994), "The complete classification of the non-Desarguesian André planes of order 25", Journal of South China Normal University, 3: 122–127
5. Dover, Jeremy M. (2019-02-27). "A genealogy of the translation planes of order 25". arXiv:1902.07838 [math.CO].
6. "Projective Planes of Order 27". ericmoorhouse.org. Retrieved 2020-11-08.