Hughes plane
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In mathematics, a Hughes plane is one of the non-Desarguesian projective planes found by (Hughes 1957). There are examples of order p2n for every odd prime p and every positive integer n.
Construction
The construction of a Hughes plane is based on a nearfield N of order p2n for p an odd prime whose kernel K has order pn and coincides with the center of N.
Properties
A Hughes plane H:[1]
- is a non-Desarguesian projective plane of odd square prime power order of Lenz-Barlotti type I.1,
- has a Desarguesian Baer subplane H0,
- is a self-dual plane in which every orthogonal polarity of H0 can be extended to a polarity of H,
- every central collineation of H0 extends to a central collineation of H, and
- the full collineation group of H has two point orbits (one of which is H0), two line orbits, and four flag orbits.
The smallest Hughes Plane (order 9)
The Hughes plane of order 9 was actually found earlier by Veblen and Wedderburn in 1907.[2] A construction of this plane can be found in (Room Kirkpatrick) where it is called the plane Ψ.
Notes
- ↑ Dembowski 1968, pg.247
- ↑ Veblen, O.; Wedderburn, J.H.M. (1907), "Non-Desarguesian and non-Pascalian geometries", Transactions of the American Mathematical Society 8 (3): 379–388, doi:10.1090/s0002-9947-1907-1500792-1, https://zenodo.org/record/1431565/files/article.pdf
References
- Dembowski, P. (1968), Finite Geometries, Berlin: Springer-Verlag
- Hughes, D. R. (1957), "A class of non-Desarguesian projective planes", Canadian Journal of Mathematics 9: 378–388, doi:10.4153/CJM-1957-045-0, ISSN 0008-414X
- Room, T. G.; Kirkpatrick, P. B. (1971). Miniquaternion geometry; an introduction to the study of projective planes. Cambridge [England]: University Press. ISBN 0-521-07926-8. OCLC 111943.
- Weibel, Charles (2007), "Survey of Non-Desarguesian Planes", Notices of the AMS 54 (10): 1294–1303, https://www.ams.org/notices/200710/
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