# 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 a Hall quasifield (also called a Hall system), H of order p2n for p a prime. The construction of the plane is the standard construction based on a quasifield (see Quasifield#Projective planes for the details.).

To build a Hall quasifield, start with a Galois field, ${\displaystyle F=GF(p^{n})}$ for p a prime and a quadratic irreducible polynomial ${\displaystyle f(x)=x^{2}-rx-s}$ over F. Extend H = F × 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 π of order n2 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 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 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 ${\displaystyle \ell }$ at a point of D (restricted to A) and the Baer subplanes that belong to D (restricted to A). D(A) is an affine plane of order n2 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 smallest Hall plane (order 9)

The Hall plane of order 9 was actually found earlier by Veblen and 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.

## 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 & Wedderburn (1907)
7. ^ Stevenson (1972, pp. 333–334)
8. ^ Hughes & Piper (1973, pg. 186)