Subplanes

The sub constructor allows subplanes of a projective or affine plane to be created. For classical planes, the SubfieldSubplane function is also provided.

sub<P | L>: Plane, List -> Plane

Given a plane \(P\), construct the subplane of \(P\) generated by the points specified by \(L\), where \(L\) is a list of one or more items of the following types:

(a)

A point of \(P\);

(b)

A set or sequence of points of \(P\);

(c)

A subplane of \(P\);

(d)

A set or sequence of subplanes of \(P\).

The set \(S\) of points defined by the list \(L\) must include a quadrangle if \(P\) is a projective plane and three non-collinear points if \(P\) is an affine plane. The function returns the smallest subplane of \(P\) containing \(S\).

SubfieldSubplane(P, F): Plane, FldFin -> Plane, PlanePtSet, PlaneLnSet

The plane obtained from the classical plane \(P\) by taking only those points of \(P\) which have all coordinates lying in \(F\), where \(F\) must be a subfield of Field(P).

Example: sub (ex-6d2e99)

In the plane \(PG_2(4)\), the points \((1 : 0 : 0), (0 : 1 : 0), (0 : 0 : 1)\) and \((1 : w : 1)\), where \(w\) is a primitive element of \({\bf F}_{4}\), form a quadrangle. We form the subplane of \(PG_2(4)\) generated by this quadrangle.

> K<w> := GF(4);
> P, V, L := FiniteProjectivePlane(K);
> S := sub< P | [ V | [1, 0, 0], [0, 1, 0], [0, 0, 1], [1, w, 1] ] >;
> S: Maximal;
Projective Plane of order 2
Points: {@ ( 1 : 0 : 0 ), ( 0 : 1 : 0 ), ( 0 : 0 : 1 ), ( 1 : w : 0 ),
( 1 : 0 : 1 ), ( 1 : w : 1 ), ( 0 : 1 : w^2 ) @}
Lines:
    {( 0 : 1 : 0 ), ( 0 : 0 : 1 ), ( 0 : 1 : w^2 )},
    {( 1 : 0 : 0 ), ( 0 : 0 : 1 ), ( 1 : 0 : 1 )},
    {( 1 : 0 : 0 ), ( 0 : 1 : 0 ), ( 1 : w : 0 )},
    {( 1 : 0 : 0 ), ( 1 : w : 1 ), ( 0 : 1 : w^2 )},
    {( 0 : 1 : 0 ), ( 1 : 0 : 1 ), ( 1 : w : 1 )},
    {( 0 : 0 : 1 ), ( 1 : w : 0 ), ( 1 : w : 1 )},
    {( 1 : w : 0 ), ( 1 : 0 : 1 ), ( 0 : 1 : w^2 )}

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We next form the subplane of \(AG_2(4)\) over \({\bf F}_{2}\).

> A := FiniteAffinePlane(4);
> S := SubfieldSubplane(A, GF(2));
> S: Maximal;
Affine Plane AG(2, 2)
> S subset A;
true

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