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Hall plane

## Summary

In mathematics, a Hall plane is a non-Desarguesian projective plane constructed by Marshall Hall Jr. (1943).[1] There are examples of order p2n for every prime p and every positive integer n provided p2n > 4.[2]

## Algebraic construction via Hall systems

The original construction of Hall planes was based on the Hall quasifield (also called a Hall system), H of order ${\displaystyle p^{2n}}$  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, ${\displaystyle F=\operatorname {GF} (p^{n})}$  for p a prime and a quadratic irreducible polynomial ${\displaystyle f(x)=x^{2}-rx-s}$  over F. Extend ${\displaystyle H=F\times F}$ , a two-dimensional vector space over F, to a quasifield by defining a multiplication on the vectors by ${\displaystyle (a,b)\circ (c,d)=(ac-bd^{-1}f(c),ad-bc+br)}$  when ${\displaystyle d\neq 0}$  and ${\displaystyle (a,b)\circ (c,0)=(ac,bc)}$  otherwise.

Writing the elements of H in terms of a basis <1, λ>, that is, identifying (x,y) with x  +  λy as x and y vary over F, we can identify the elements of F as the ordered pairs (x, 0), i.e. 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 f(α) =  0;
2. F is in the kernel of H (meaning that (α  +  β)c  =  αc  +  βc, and (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.[3]

## Derivation

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.[4] Start with a projective plane ${\displaystyle \pi }$  of order ${\displaystyle n^{2}}$  and designate one line ${\displaystyle \ell }$  as its line at infinity. Let A be the affine plane ${\displaystyle \pi \setminus \ell }$ . A set D of ${\displaystyle n+1}$  points of ${\displaystyle \ell }$  is called a derivation set if for every pair of distinct points X and Y of A which determine a line meeting ${\displaystyle \ell }$  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 ${\displaystyle \operatorname {D} (A)}$  as follows: The points of ${\displaystyle \operatorname {D} (A)}$  are the points of A. The lines of ${\displaystyle \operatorname {D} (A)}$  are the lines of ${\displaystyle \pi }$  which do not meet ${\displaystyle \ell }$  at a point of D (restricted to A) and the Baer subplanes that belong to D (restricted to A). The set ${\displaystyle \operatorname {D} (A)}$  is an affine plane of order ${\displaystyle n^{2}}$  and it, or its projective completion, is called a derived plane.[5]

## Properties

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.

## The Hall plane of order 9

Hall plane of order 9
Order9
Lenz-Barlotti ClassIVa.3
Automorphisms${\displaystyle 2^{8}\times 3^{5}\times 5}$
Point Orbit Lengths10, 81
Line Orbit Lengths1, 90
PropertiesTranslation 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.

### Construction

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.[6] 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 ${\displaystyle f(x)=x^{2}+1}$ , ${\displaystyle g(x)=x^{2}-x-1}$  or ${\displaystyle h(x)=x^{2}+x-1}$ .[7] The first of these produces an associative quasifield,[8] 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.

### Properties

#### Automorphism Group

The Hall plane of order 9 is the unique projective plane, finite or infinite, which has Lenz-Barlotti class IVa.3.[9] 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 ${\displaystyle S_{5}}$  on these 5 pairs.[10]

#### Unitals

The Hall plane of order 9 admits four inequivalent embedded unitals.[11] Two of these unitals arise from Buekenhout's[12] 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[13] to also be embeddable in the dual Hall plane. Another of the unitals arises from the construction of Barlotti and Lunardon.[14] The fourth has an automorphism group of order 8 isomorphic to the quaternions, and is not part of any known infinite family.

## Notes

1. ^ Hall Jr. (1943)
2. ^ Although the constructions will provide a projective plane of order 4, the unique such plane is Desarguesian and is generally not considered to be a Hall plane.
3. ^ Hughes & Piper (1973, pg. 183)
4. ^ Hughes & Piper (1973, pp. 202–218, Chapter X. Derivation)
5. ^ Hughes & Piper (1973, pg. 203, Theorem 10.2)
6. ^ Veblen, Oscar; Wedderburn, Joseph H.M. (1907), "Non-Desarguesian and non-Pascalian geometries" (PDF), Transactions of the American Mathematical Society, 8: 379–388, doi:10.2307/1988781
7. ^ Stevenson, Frederick W. (1972), Projective Planes, San Francisco: W.H. Freeman and Company, ISBN 0-7167-0443-9 Pages 333–334.
8. ^ D. Hughes and F. Piper (1973). Projective Planes. Springer-Verlag. ISBN 0-387-90044-6. Page 186.
9. ^ Dembowski, Peter (1968). Finite Geometries : Reprint of the 1968 Edition. Berlin, Heidelberg: Springer Berlin Heidelberg. ISBN 978-3-642-62012-6. OCLC 851794158. Page 126.
10. ^ André, Johannes (1955-12-01). "Projektive Ebenen über Fastkörpern". Mathematische Zeitschrift (in German). 62 (1): 137–160. doi:10.1007/BF01180628. ISSN 1432-1823.
11. ^ Penttila, Tim; Royle, Gordon F. (1995-11-01). "Sets of type (m, n) in the affine and projective planes of order nine". Designs, Codes and Cryptography. 6 (3): 229–245. doi:10.1007/BF01388477. ISSN 1573-7586.
12. ^ Buekenhout, F. (July 1976). "Existence of unitals in finite translation planes of order ${\displaystyle q^{2}}$  with a kernel of order ${\displaystyle q}$ ". Geometriae Dedicata. 5 (2). doi:10.1007/BF00145956. ISSN 0046-5755.
13. ^ Grüning, Klaus (1987-06-01). "A class of unitals of order ${\displaystyle q}$  which can be embedded in two different planes of order ${\displaystyle q^{2}}$ ". Journal of Geometry. 29 (1): 61–77. doi:10.1007/BF01234988. ISSN 1420-8997.
14. ^ Barlotti, A.; Lunardon, G. (1979). "Una classe di unitals nei ${\displaystyle \Delta }$ -piani". Rivisita di Matematica della Università di Parma. 4: 781–785.