Editing Projective plane (section)

==Subplanes==

A '''subplane''' of a projective plane <math>({\mathcal P},{\mathcal L},I)</math> is a pair of subsets <math>({\mathcal P'},{\mathcal L'})</math> where <math>{\mathcal P'}\subseteq{\mathcal P}</math>,
<math>{\mathcal L'}\subseteq{\mathcal L}</math> and <math>({\mathcal P'},{\mathcal L'},I')</math> is itself a projective plane with respect to the restriction <math>I'</math> of the incidence relation <math>I</math> to <math>({\mathcal P'}\cup{\mathcal L'})\times({\mathcal P'}\cup{\mathcal L'})</math>.

{{harv|Bruck|1955}} proves the following theorem. Let Π be a finite projective plane of order ''N'' with a proper subplane Π<sub>0</sub> of order ''M''. Then either ''N'' = ''M''<sup>2</sup> or ''N'' ≥ ''M''<sup>2</sup> + ''M''.

A subplane <math>({\mathcal P'},{\mathcal L'})</math> of <math>({\mathcal P},{\mathcal L},I)</math> is a '''Baer subplane''' if every line in <math>{\mathcal L}\setminus{\mathcal L'}</math> is incident with exactly one point in <math>\mathcal P'</math> and every point in <math>{\mathcal P}\setminus{\mathcal P'}</math> is incident with exactly one line of <math>\mathcal L'</math>.

A finite Desarguesian projective plane of order
<math>q</math> admits Baer subplanes (all necessarily Desarguesian) if and
only if <math>q</math> is square; in this
case the order of the Baer subplanes is <math>\sqrt{q}</math>.

In the finite Desarguesian planes PG(2,&nbsp;''p<sup>n</sup>''), the subplanes have orders which are the orders of the subfields of the finite field GF(''p<sup>n</sup>''), that is, ''p<sup>i</sup>'' where ''i'' is a divisor of ''n''. In non-Desarguesian planes however, Bruck's theorem gives the only information about subplane orders. The case of equality in the inequality of this theorem is not known to occur. Whether or not there exists a subplane of order ''M'' in a plane of order ''N'' with ''M''<sup>2</sup> + ''M'' = ''N'' is an open question. If such subplanes existed there would be projective planes of composite (non-prime power) order.

===Fano subplanes===

A '''Fano subplane''' is a subplane isomorphic to PG(2,&nbsp;2), the unique projective plane of order 2.

If you consider a ''[[quadrangle]]'' (a set of 4 points no three collinear) in this plane, the points determine six of the lines of the plane. The remaining three points (called the ''diagonal points'' of the quadrangle) are the points where the lines that do not intersect at a point of the quadrangle meet. The seventh line consists of all the diagonal points (usually drawn as a circle or semicircle).

In finite desarguesian planes, PG(2,&nbsp;''q''), Fano subplanes exist if and only if ''q'' is even (that is, a power of 2). The situation in non-desarguesian planes is unsettled. They could exist in any non-desarguesian plane of order greater than 6, and indeed, they have been found in all non-desarguesian planes in which they have been looked for (in both odd and even orders).

An open question, apparently due to [[Hanna Neumann]] though not published by her, is: Does every non-desarguesian plane contain a Fano subplane?

A theorem concerning Fano subplanes due to {{harv|Gleason|1956}} is:

:If every quadrangle in a finite projective plane has collinear diagonal points, then the plane is desarguesian (of even order).