Hughes plane

For the large airplane built by Hughes, see Spruce goose.

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. 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. A construction of this plane can be found in Room & Kirkpatrick (1971) where it is called the plane Ψ.

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 ISBN 0-521-07926-8 .

• Dembowski 1968, pg.247