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In mathematics, a Hughes plane is one of the
non-Desarguesian projective plane In mathematics, a non-Desarguesian plane is a projective plane that does not satisfy Desargues' theorem (named after Girard Desargues), or in other words a plane that is not a Desarguesian plane. The theorem of Desargues is true in all projective ...
s found by . There are examples of order ''p''2''n'' 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 ''p''''2n'' for ''p'' an odd prime whose kernel K has order ''p''''n'' and coincides with the center of N.


Properties

A Hughes plane H: # 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. A construction of this plane can be found in where it is called the plane Ψ.


Notes


References

* * * T. G. Room & P.B. Kirkpatrick (1971) ''Miniquaternion geometry'', Part III Miniquaternion planes, chapter V The Plane Ψ, pp 130–68,
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Pres ...
. *{{Citation , last1=Weibel , first1=Charles , title=Survey of Non-Desarguesian Planes , url=https://www.ams.org/notices/200710/ , year=2007 , journal= Notices of the AMS , volume= 54 , issue=10 , pages=1294–1303 Projective geometry Finite geometry