Hall plane

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]

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]

1. Hall planes are translation planes.
2. The Hall plane of order 9 is the only projective plane of Lenz-Barlotti type IVa.3, finite or infinite.[6] All other Hall planes are of Lenz-Barlotti type IVa.1.
3. All finite Hall planes of the same order are isomorphic.
4. Hall planes are not self-dual.
5. All finite Hall planes contain subplanes of order 2 (Fano subplanes).
6. All finite Hall planes contain subplanes of order different from 2.
7. Hall planes are André planes.

The Hall plane of order 9 was actually found earlier by Oswald Veblen and Joseph Wedderburn in 1907.[7] 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 ${\displaystyle h(x)=x^{2}+x-1}$. [8] The first of these produces an associative quasifield,[9] 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.

• 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