# 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 *p*^{2n} 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**:^{[1]}

- is a non-Desarguesian projective plane of odd square prime power order of Lenz-Barlotti type I.1,
- has a Desarguesian Baer subplane
**H**_{0}, - is a self-dual plane in which every orthogonal polarity of
**H**_{0}can be extended to a polarity of**H**, - every central collineation of
**H**_{0}extends to a central collineation of**H**, and - the full collineation group of
**H**has two point orbits (one of which is**H**_{0}), 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

- ↑ Dembowski 1968, pg.247
- ↑ 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

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