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

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]

Order	Number of non-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

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.
7. Dover, Jeremy M. (2021-05-16). "Computational Enumeration of Andrè Planes". arXiv:2105.07439 [math.CO].