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 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.[2]

Properties

  1. André planes exist for all proper prime powers with prime and a positive integer greater than one.
  2. 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.[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].
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