Hughes plane

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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[edit]

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[edit]

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)[edit]

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[edit]

  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[edit]