Hall plane

The original construction of Hall planes was based on the Hall quasifield (also called a Hall system), H of order p²ⁿ for p a prime. The creation of the plane from the quasifield follows the standard construction (see quasifield for details).

To build a Hall quasifield, start with a Galois field, {{nowrap|1=F = GF(pⁿ)}} for p a prime and a quadratic irreducible polynomial {{nowrap|1=f(x) = x² − rx − s}} over F. Extend {{nowrap|1=H = F × F}}, a two-dimensional vector space over F, to a quasifield by defining a multiplication on the vectors by {{nowrap|1=(a, b) ∘ (c, d) = (ac − bd⁻¹f(c), ad − bc + br)}} when {{nowrap|d ≠ 0}} and {{nowrap|1=(a, b) ∘ (c, 0) = (ac, bc)}} otherwise.

Writing the elements of H in terms of a basis {{angbr|1, λ}}, that is, identifying {{nowrap|(x, y)}} with {{nowrap|x + λy}} as x and y vary over F, we can identify the elements of F as the ordered pairs {{nowrap|(x, 0)}}, i.e. {{nowrap|x + λ0}}. The properties of the defined multiplication which turn the right vector space H into a quasifield are:

1. every element α of H not in F satisfies the quadratic equation {{nowrap|1=f(α) = 0}};
2. F is in the kernel of H (meaning that {{nowrap|1=(α + β)c = αc + βc}}, and {{nowrap|1=(αβ)c = α(βc)}} for all α, β in H and all c in F); and
3. every element of F commutes (multiplicatively) with all the elements of H.{{harvtxt|Hughes|Piper|1973|p=183}}

Another construction that produces Hall planes is obtained by applying derivation to Desarguesian planes.

A process, due to T. G. Ostrom, which replaces certain sets of lines in a projective plane by alternate sets in such a way that the new structure is still a projective plane is called derivation. We give the details of this process.{{harvtxt|Hughes|Piper|1973|pp=202–218|loc=Chapter X. Derivation}} Start with a projective plane π of order n² and designate one line ℓ as its line at infinity. Let A be the affine plane {{nowrap|π ∖ ℓ}}. A set D of {{nowrap|n + 1}} points of ℓ is called a derivation set if for every pair of distinct points X and Y of A which determine a line meeting ℓ in a point of D, there is a Baer subplane containing X, Y and D (we say that such Baer subplanes belong to D.) Define a new affine plane D(A) as follows: The points of D(A) are the points of A. The lines of D(A) are the lines of π which do not meet ℓ at a point of D (restricted to A) and the Baer subplanes that belong to D (restricted to A). The set D(A) is an affine plane of order n² and it, or its projective completion, is called a derived plane.{{harvtxt|Hughes|Piper|1973|p=203|loc=Theorem 10.2}}

1. Hall planes are translation planes.
2. All finite Hall planes of the same order are isomorphic.
3. Hall planes are not self-dual.
4. All finite Hall planes contain subplanes of order 2 (Fano subplanes).
5. All finite Hall planes contain subplanes of order different from 2.
6. Hall planes are André planes.

{{Infobox finite projective plane

|name=Hall plane of order 9

|order=9

|lbclass=IVa.3

|automorphisms=2⁸ × 3⁵ × 5

|points=10, 81

|lines=1, 90

|properties=Translation plane

}}

The Hall plane of order 9 is the smallest Hall plane, and one of the three smallest examples of a finite non-Desarguesian projective plane, along with its dual and the Hughes plane of order 9.{{citation |last1=Moorhouse |first1=G. Eric |title=Projective Planes of Small Order |url=https://ericmoorhouse.org/pub/planes/ |year=2017 }} explicitly lists the incidence structure of these planes.

While usually constructed in the same way as other Hall planes, the Hall plane of order 9 was actually found earlier by Oswald Veblen and Joseph Wedderburn in 1907.{{citation |last1=Veblen |first1=Oswald |title=Non-Desarguesian and non-Pascalian geometries |url=http://www.ams.org/tran/1907-008-03/S0002-9947-1907-1500792-1/S0002-9947-1907-1500792-1.pdf |journal=Transactions of the American Mathematical Society |volume=8 |pages=379–388 |year=1907 |doi=10.2307/1988781 |last2=Wedderburn |first2=Joseph H.M. |issue=3 |jstor=1988781 |authorlink1=Oswald Veblen|authorlink2=Joseph Wedderburn|doi-access=free}} There are four quasifields of order nine which can be used to construct the Hall plane of order nine. Three of these are Hall systems generated by the irreducible polynomials {{nowrap|1=f(x) = x² + 1}}, {{nowrap|1=g(x) = x² − x − 1}} or {{nowrap|1=h(x) = x² + x − 1}}.{{citation |last=Stevenson |first=Frederick W. |title=Projective Planes |year=1972 |place=San Francisco |publisher=W.H. Freeman and Company |isbn=0-7167-0443-9 |pages=333–334}} The first of these produces an associative quasifield,{{cite book |author=D. Hughes and F. Piper |title=Projective Planes |publisher=Springer-Verlag |year=1973 |isbn=0-387-90044-6 |page=186 }} that is, a near-field, and it was in this context that the plane was discovered by Veblen and Wedderburn. This plane is often referred to as the nearfield plane of order nine.

The Hall plane of order 9 is the unique projective plane, finite or infinite, which has Lenz–Barlotti class IVa.3.{{cite book |last=Dembowski |first=Peter |url=https://www.worldcat.org/oclc/851794158 |title=Finite Geometries : Reprint of the 1968 Edition |date=1968 |publisher=Springer Berlin Heidelberg |isbn=978-3-642-62012-6 |location=Berlin, Heidelberg |oclc=851794158 |page=126 }} Its automorphism group acts on its (necessarily unique) translation line imprimitively, having 5 pairs of points that the group preserves set-wise; the automorphism group acts as S₅ on these 5 pairs.{{cite journal |last=André |first=Johannes |date=1955-12-01 |title=Projektive Ebenen über Fastkörpern |url=https://doi.org/10.1007/BF01180628 |journal=Mathematische Zeitschrift |language=de |volume=62 |issue=1 |pages=137–160 |doi=10.1007/BF01180628|s2cid=122641224 |issn=1432-1823|url-access=subscription }}

The Hall plane of order 9 admits four inequivalent embedded unitals.{{cite journal |last1=Penttila |first1=Tim |last2=Royle |first2=Gordon F. |date=1995-11-01 |title=Sets of type (m, n) in the affine and projective planes of order nine |url=https://doi.org/10.1007/BF01388477 |journal=Designs, Codes and Cryptography |language=en |volume=6 |issue=3 |pages=229–245 |doi=10.1007/BF01388477|s2cid=43638589 |issn=1573-7586 |url-access=subscription }} Two of these unitals arise from Buekenhout's{{cite journal |last=Buekenhout |first=F. |date=July 1976 |title=Existence of unitals in finite translation planes of order q² with a kernel of order q |url=http://link.springer.com/10.1007/BF00145956 |journal=Geometriae Dedicata |language=en |volume=5 |issue=2 |doi=10.1007/BF00145956 |s2cid=123037502 |issn=0046-5755 |url-access=subscription }} constructions: one is parabolic, meeting the translation line in a single point, while the other is hyperbolic, meeting the translation line in 4 points. The latter of these two unitals was shown by Grüning{{Cite journal |last=Grüning |first=Klaus |date=1987-06-01 |title=A class of unitals of order q which can be embedded in two different planes of order q² |url=https://doi.org/10.1007/BF01234988 |journal=Journal of Geometry |language=en |volume=29 |issue=1 |pages=61–77 |doi=10.1007/BF01234988 |s2cid=117872040 |issn=1420-8997 |url-access=subscription }} to also be embeddable in the dual Hall plane. Another of the unitals arises from the construction of Barlotti and Lunardon.{{cite journal |last1=Barlotti |first1=A. |last2=Lunardon |first2=G. |date=1979 |title=Una classe di unitals nei Δ-piani |journal=Rivisita di Matematica della Università di Parma |volume=4 |pages=781–785 }} The fourth has an automorphism group of order 8 isomorphic to the quaternions, and is not part of any known infinite family.

{{reflist}}

{{refbegin|30em}}

• {{citation |last=Dembowski |first=P. |title=Finite Geometries |year=1968 |publisher=Springer-Verlag |place=Berlin }}
• {{citation |last=Hall |first=Marshall Jr. |authorlink=Marshall Hall (mathematician) |title=Projective Planes |url=https://www.ams.org/journals/tran/1943-054-02/S0002-9947-1943-0008892-4/S0002-9947-1943-0008892-4.pdf |year=1943 |journal=Transactions of the American Mathematical Society |volume=54 |issue=2 | jstor=1990331 | mr=0008892 | issn=0002-9947 |pages=229–277 | doi=10.2307/1990331 |doi-access=free }}
• {{cite book | first1=D. |last1=Hughes |first2=F. |last2=Piper | title=Projective Planes | publisher=Springer-Verlag | year=1973 | isbn=0-387-90044-6 }}
• {{citation | last = Stevenson | first = Frederick W. | title = Projective Planes | publisher = W.H. Freeman and Company | place = San Francisco |year = 1972 | isbn = 0-7167-0443-9 }}
• {{citation |last1=Veblen |first1=Oswald |authorlink1=Oswald Veblen |last2=Wedderburn|first2=Joseph H.M.|authorlink2=Joseph Wedderburn|title=Non-Desarguesian and non-Pascalian geometries |url=https://www.ams.org/tran/1907-008-03/S0002-9947-1907-1500792-1/S0002-9947-1907-1500792-1.pdf |journal=Transactions of the American Mathematical Society |year=1907 |volume=8 |issue=3 |pages=379–388|doi=10.2307/1988781|jstor=1988781 |doi-access=free}}
• {{citation | last1=Weibel | first1=Charles | authorlink=Charles Weibel | title=Survey of Non-Desarguesian Planes | url=https://www.ams.org/notices/200710/tx071001294p.pdf | year=2007 | journal= Notices of the American Mathematical Society | volume= 54 | issue=10 | pages=1294–1303}}

{{refend}}

Category:Projective geometry

Category:Finite geometry