Hughes plane
# Hughes plane

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

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

