Hughes plane
{{for|the large airplane built by Hughes|Spruce goose}}
In mathematics, a Hughes plane is one of the non-Desarguesian projective planes found by {{harvtxt|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:{{harvnb|Dembowski|1968|loc=pg.247}}
- is a non-Desarguesian projective plane of odd square prime power order of Lenz-Barlotti type I.1,
- has a Desarguesian Baer subplane H0,
- is a self-dual plane in which every orthogonal polarity of H0 can be extended to a polarity of H,
- every central collineation of H0 extends to a central collineation of H, and
- 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.{{Citation|last1=Veblen|first1=O.|last2=Wedderburn|first2=J.H.M.|title=Non-Desarguesian and non-Pascalian geometries|journal=Transactions of the American Mathematical Society|year=1907|volume=8|issue=3|pages=379–388|doi=10.1090/s0002-9947-1907-1500792-1|url=https://zenodo.org/record/1431565/files/article.pdf|doi-access=free}} A construction of this plane can be found in {{harvtxt|Room|Kirkpatrick|1971}} where it is called the plane Ψ.
Notes
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References
- {{Citation|last=Dembowski|first=P.|authorlink=Peter Dembowski|title=Finite Geometries|year=1968|publisher=Springer-Verlag|place=Berlin}}
- {{Citation | doi=10.4153/CJM-1957-045-0 | last1=Hughes | first1=D. R. | title=A class of non-Desarguesian projective planes |mr=0087960 | year=1957 | journal=Canadian Journal of Mathematics | issn=0008-414X | volume=9 | pages=378–388 }}
- {{cite book | last=Room | first=T. G. | last2=Kirkpatrick | first2=P. B. | title=Miniquaternion geometry; an introduction to the study of projective planes | publisher=University Press | publication-place=Cambridge [England] | date=1971 | isbn=0-521-07926-8 | oclc=111943}}
- {{Citation | last1=Weibel | first1=Charles | title=Survey of Non-Desarguesian Planes | url=https://www.ams.org/notices/200710/ | year=2007 | journal= Notices of the AMS | volume= 54 | issue=10 | pages=1294–1303}}