Nearfield Planes

A nearfield \(N\) is said to be planar if the mapping \(x\mapsto -xa+xb\) is a permutation of \(N\) whenever \(a\ne b\). Every finite nearfield is planar.

Given a finite nearfield \(N\), there is an affine plane \(\cal A\) with point set \(N\times N\) and lines given by the equations

\[\begin{split}\begin{aligned}y &= xm+b\\ x &= c\end{aligned}\end{split}\]

Let \(\cal P\) be the corresponding projective plane, obtained from \(\cal A\) by adjoining a line \(L_\infty\) called the line at infinity. We label the points of \(\cal P\) with triples of elements of \(N\) as follows.

(1) For every point \((x,y)\) of \(\cal A\) there is a point \([1,x,y]\) of \(\cal P\).

(2) For every \(m\) there is an “ideal” point \([0,1,m]\) of \(\cal P\) which lies on every line \(y = xm+b\) (\(b\in N\)) and on \(L_\infty\).

(3)There is a point \([0,0,1]\) of \(\cal P\) which lies on every line \(x = c\) and on \(L_\infty\).

The lines of \(\cal P\) may also be labelled by triples of elements of \(N\): the line \(y = xm+b\) corresponds to the triple \([-b,-m,1]\) and the line \(x = c\) corresponds to \([-c,1,0]\). The line \(L_\infty\) is labelled \([1,0,0]\). A point \(\pi = [w,x,y]\) is incident with a line \(L = [a,b,c]\) if and only if \(wa+xb+yc= 0\).

Every collineation of \(\cal A\) extends to a collineation of \(\cal P\).

ProjectivePlane(N : parameters): Nfd -> PlaneProj, PlanePtSet, PlaneLnSet

Check: BoolElt                    Default: false

The finite projective plane coordinatised by the nearfield \(N\). The points of the nearfield plane are represented as triples of Galois field elements.

Example: projplane (ex-f4d8e0)

> N := DicksonNearfield(3,2);
> pl := ProjectivePlane(N);
> A := AutomorphismGroup(pl);
> #A;
311040
> CompositionFactors(A);
    G
    |  Cyclic(2)
    *
    |  Alternating(5)
    *
    |  Cyclic(2)
    *
    |  Cyclic(2)
    *
    |  Cyclic(2)
    *
    |  Cyclic(2)
    *
    |  Cyclic(2)
    *
    |  Cyclic(3)
    *
    |  Cyclic(3)
    *
    |  Cyclic(3)
    *
    |  Cyclic(3)
    1

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Hughes Planes

In 1957 Hughes [Hughes, 1957] discovered a class of finite projective planes constructed from the Dickson nearfields which have rank 2 over their kernel. Neither these planes nor their duals are translation planes and therefore they cannot be obtained by the coordinatisation method of the previous section. Hughes’ methods required the kernel to be central but in 1960 the construction was generalised by Rosati [Rosati, 1960] to include the Zassenhaus nearfields (see also Dembowski [Dembowski, 1968, §5.4] and [Dembowski, 1971]). For simplicity of notation we shall use the term ‘Hughes plane’ to include both Hughes planes and generalised Hughes planes.

HughesPlane(N : parameters): Nfd -> PlaneProj, PlanePtSet, PlaneLnSet

Check: BoolElt                    Default: false

The Hughes plane based on the nearfield \(N\).

Example: hughes (ex-c201b6)

We construct the Desarguesian projective plane PG(2,49) and then, using nearfields of order 49, we construct three non-Desarguesian projective planes. These four planes can be distinguished by the orders of their collineation groups.

> DP := FiniteProjectivePlane(49); // Desarguesian plane
> DP;
Projective Plane PG(2, 49)
> CD  := CollineationGroup(DP);
> FactoredOrder(CD);
[ <2, 10>, <3, 3>, <5, 2>, <7, 6>, <19, 1>, <43, 1> ]
> N := DicksonNearfield(7,2);
> NP := ProjectivePlane(N);
> NP;
Projective Plane of order 49
> CN := CollineationGroup(NP);
> FactoredOrder(CN);
[ <2, 10>, <3, 2>, <7, 4> ]
> Z := ZassenhausNearfield(3);
> #Z;
49
> ZP := ProjectivePlane(Z);
> CZ := CollineationGroup(ZP);
> FactoredOrder(CZ);
[ <2, 9>, <3, 3>, <7, 4> ]
> HP := HughesPlane(N);
> HP;
Projective Plane of order 49
> CH := CollineationGroup(HP);
> FactoredOrder(CH);
[ <2, 6>, <3, 3>, <7, 3>, <19, 1> ]
> CompositionFactors(CH);
    G
    |  Cyclic(3)
    *
    |  A(2, 7)                = L(3, 7)
    *
    |  Cyclic(2)
    1

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