Hall plane

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
Order	9
Lenz-Barlotti Class	IVa.3
Automorphisms	${\displaystyle 2^{8}\times 3^{5}\times 5}$
Point Orbit Lengths	10, 81
Line Orbit Lengths	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.

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[]

References[]

• Dembowski, P. (1968), Finite Geometries, Berlin: Springer-Verlag
• Hall Jr., Marshall (1943), "Projective Planes" (PDF), Transactions of the American Mathematical Society, 54: 229–277, doi:10.2307/1990331, ISSN 0002-9947, JSTOR 1990331, MR 0008892
• D. Hughes and F. Piper (1973). Projective Planes. Springer-Verlag. ISBN 0-387-90044-6.
• Stevenson, Frederick W. (1972), Projective Planes, San Francisco: W.H. Freeman and Company, ISBN 0-7167-0443-9
• 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
• Weibel, Charles (2007), "Survey of Non-Desarguesian Planes" (PDF), Notices of the American Mathematical Society, 54 (10): 1294–1303