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