Editing Projective plane (section)

===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).