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 Plane_sub (H150E4)

In the plane PG2(4), the points (1 : 0 : 0), (0 : 1 : 0), (0 : 0 : 1) and (1 : w : 1), where w is a primitive element of GF(4), form a quadrangle. We form the subplane of PG2(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 )}
We next form the subplane of AG2(4) over GF(2). 
> A := FiniteAffinePlane(4);
> S := SubfieldSubplane(A, GF(2));
> S: Maximal;
Affine Plane AG(2, 2)
> S subset A;
true
V2.28, 13 July 2023