Editing Projective space (section)

=== Finite projective spaces and planes ===
{{details|topic=finite projective planes|Projective plane#Finite projective planes}} 
[[Image:Fano plane.svg|thumb|right|The [[Fano plane]] ]]

A ''finite projective space'' is a projective space where {{math|''P''}} is a finite set of points. In any finite projective space, each line contains the same number of points and the ''order'' of the space is defined as one less than this common number. For finite projective spaces of dimension at least three, [[Wedderburn's little theorem|Wedderburn's theorem]] implies that the division ring over which the projective space is defined must be a [[finite field]], {{math|GF(''q'')}}, whose order (that is, number of elements) is {{math|''q''}} (a prime power). A finite projective space defined over such a finite field has {{math|''q'' + 1}} points on a line, so the two concepts of order coincide. Notationally, {{math|PG(''n'', GF(''q''))}} is usually written as {{math|PG(''n'', ''q'')}}.

All finite fields of the same order are isomorphic, so, up to isomorphism, there is only one finite projective space for each dimension greater than or equal to three, over a given finite field. However, in dimension two there are non-Desarguesian planes. Up to isomorphism there are
{{block indent | em = 1.5 | text = 1, 1, 1, 1, 0, 1, 1, 4, 0, ... {{OEIS|id=A001231}}}}
finite projective planes of orders 2, 3, 4, ..., 10, respectively. The numbers beyond this are very difficult to calculate and are not determined except for some zero values due to the [[Bruck–Ryser–Chowla theorem|Bruck–Ryser theorem]].

The smallest projective plane is the [[Fano plane]], {{math|PG(2, 2)}} with 7 points and 7 lines. The smallest 3-dimensional projective space is [[PG(3,2)|{{math|PG(3, 2)}}]], with 15 points, 35 lines and 15 planes.