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 [math]\displaystyle{ F = GF(q) }[/math] be a finite field, and let [math]\displaystyle{ K = GF(q^n) }[/math] be a degree [math]\displaystyle{ n }[/math] extension field of [math]\displaystyle{ F }[/math]. Let [math]\displaystyle{ \Gamma }[/math] be the group of field automorphisms of [math]\displaystyle{ K }[/math] over [math]\displaystyle{ F }[/math], and let [math]\displaystyle{ \beta }[/math] be an arbitrary mapping from [math]\displaystyle{ F }[/math] to [math]\displaystyle{ \Gamma }[/math] such that [math]\displaystyle{ \beta(1)=1 }[/math]. Finally, let [math]\displaystyle{ N }[/math] be the norm function from [math]\displaystyle{ K }[/math] to [math]\displaystyle{ F }[/math].
Define a quasifield [math]\displaystyle{ Q }[/math] with the same elements and addition as K, but with multiplication defined via [math]\displaystyle{ a \circ b = a^{\beta(N(b))} \cdot b }[/math], where [math]\displaystyle{ \cdot }[/math] denotes the normal field multiplication in [math]\displaystyle{ K }[/math]. Using this quasifield to construct a plane yields an André plane.[2]
Properties
- André planes exist for all proper prime powers [math]\displaystyle{ p^n }[/math] with [math]\displaystyle{ p }[/math] prime and [math]\displaystyle{ n }[/math] a positive integer greater than one.
- Non-Desarguesian André planes exist for all proper prime powers except for [math]\displaystyle{ 2^n }[/math] where [math]\displaystyle{ n }[/math] 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
- ↑ André, Johannes (1954), "Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe", Mathematische Zeitschrift 60: 156–186, doi:10.1007/BF01187370, ISSN 0025-5874, http://resolver.sub.uni-goettingen.de/purl?GDZPPN002384345
- ↑ Weibel, Charles (2007), "Survey of Non-Desarguesian Planes", Notices of the AMS 54 (10): 1294–1303, https://www.ams.org/notices/200710/
- ↑ "Projective Planes of Order 16". http://ericmoorhouse.org/pub/planes16/.
- ↑ Chen, G. (1994), "The complete classification of the non-Desarguesian André planes of order 25", Journal of South China Normal University 3: 122–127
- ↑ Dover, Jeremy M. (2019-02-27). "A genealogy of the translation planes of order 25". arXiv:1902.07838 [math.CO].
- ↑ "Projective Planes of Order 27". http://ericmoorhouse.org/pub/planes27/.
- ↑ Dover, Jeremy M. (2021-05-16). "Computational Enumeration of Andrè Planes". arXiv:2105.07439 [math.CO].
 |  |
