Hughes plane

In mathematics, a Hughes plane is one of the non-Desarguesian projective planes found by Daniel 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]

1. is a non-Desarguesian projective plane of odd square prime power order of Lenz-Barlotti type I.1,
2. has a Desarguesian Baer subplane H₀,
3. is a self-dual plane in which every orthogonal polarity of H₀ can be extended to a polarity of H,
4. every central collineation of H₀ extends to a central collineation of H, and
5. 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 (1971) where it is called the plane Ψ.

Notes

1. Dembowski 1968, pg.247
2. Veblen, O.; Wedderburn, J.H.M. (1907), "Non-Desarguesian and non-Pascalian geometries", Transactions of the AMS, 8: 379–388, doi:10.1090/s0002-9947-1907-1500792-1

References

• 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