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.
- 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 Room & Kirkpatrick (1971) where it is called the plane Ψ.
- Dembowski, P. (1968), Finite Geometries, Berlin: Springer-Verlag
- Hughes, D. R. (1957), "A class of non-Desarguesian projective planes", Canadian Journal of Mathematics. Journal Canadien de Mathématiques, 9: 378–388, doi:10.4153/CJM-1957-045-0, ISSN 0008-414X, MR 0087960
- T. G. Room & P.B. Kirkpatrick (1971) Miniquaternion geometry, Part III Miniquaternion planes, chapter V The Plane Ψ, pp 130–68, Cambridge University Press ISBN 0-521-07926-8 .
- Weibel, Charles (2007), "Survey of Non-Desarguesian Planes", Notices of the AMS, 54 (10): 1294–1303