Editing Projective plane (section)

==Finite projective planes==
{{projective_plane_of_order_7.svg}}
It can be shown that a projective plane has the same number of lines as it has points (infinite or finite).  Thus, for every finite projective plane there is an [[integer]] ''N'' ≥ 2 such that the plane has
:''N''<sup>2</sup> + ''N'' + 1 points,
:''N''<sup>2</sup> + ''N'' + 1 lines,
:''N'' + 1 points on each line, and
:''N'' + 1 lines through each point.
The number ''N'' is called the '''order''' of the projective plane.

The projective plane of order 2 is called the [[Fano plane]]. See also the article on [[finite geometry]].

Using the vector space construction with finite fields there exists a projective plane of order {{nowrap|1=''N'' = ''p''<sup>''n''</sup>}}, for each prime power ''p''<sup>''n''</sup>.  In fact, for all known finite projective planes, the order ''N'' is a prime power.{{cn|date=November 2024}}

The existence of finite projective planes of other orders is an open question.  The only general restriction known on the order is the [[Bruck–Ryser–Chowla theorem]] that if the order ''N'' is [[modular arithmetic|congruent]] to 1 or 2 mod 4, it must be the sum of two squares.  This rules out {{nowrap|1=''N'' = 6}}.  The next case {{nowrap|1=''N'' = 10}} has been ruled out by massive computer calculations.{{sfnp|Lam|1991}}  Nothing more is known; in particular, the question of whether there exists a finite projective plane of order {{nowrap|1=''N'' = 12}} is still open.{{cn|date=November 2024}}

Another longstanding open problem is whether there exist finite projective planes of ''prime'' order which are not finite field planes (equivalently, whether there exists a non-Desarguesian projective plane of prime order).{{cn|date=November 2024}}

A projective plane of order ''N'' is a Steiner {{nowrap|S(2, ''N'' + 1, ''N''<sup>2</sup> + ''N'' + 1)}} system
(see [[Steiner system]]).  Conversely, one can prove that all Steiner systems of this form ({{nowrap|1=''λ'' = 2}}) are projective planes.

[[Automorphism]]s for PG(''n'',''k''), with ''k''=''p''<sup>''m''</sup>, ''p''=prime is (''m''!)(''k''<sup>''n''+1</sup> − 1)(''k''<sup>''n''+1</sup> − ''k'')(''k''<sup>''n''+1</sup> − ''k''<sup>2</sup>)...(''k''<sup>''n''+1</sup> − ''k''<sup>''n''</sup>)/(''k'' − 1).

The number of mutually [[orthogonal Latin squares]] of order ''N'' is at most {{nowrap|''N'' &minus; 1}}.  {{nowrap|''N'' &minus; 1}} exist if and only if there is a projective plane of order ''N''.

While the classification of all projective planes is far from complete, results are known for small orders:
*2 : all isomorphic to [[Fano plane|PG(2, 2)]], configuration (7<sub>3</sub>) with 168 automorphisms
*3 : all isomorphic to [[PG(2,3)|PG(2, 3)]], configuration (15<sub>4</sub>) with 1516 automorphisms
*4 : all isomorphic to PG(2, 4), configuration (21<sub>5</sub>) with 120,960 automorphisms, doubled by prime power, 4=2<sup>2</sup>
*5 : all isomorphic to PG(2, 5), configuration (31<sub>6</sub>) with 372,000 automorphisms
*6 : impossible as the order of a projective plane, proved by [[Gaston Tarry|Tarry]] who showed that [[Euler]]'s [[thirty-six officers problem]] has no solution. However, the connection between these problems was not known until [[R. C. Bose|Bose]] proved it in 1938.<ref>{{harvp|Lam|1991|p=306}}. "In 1938, Bose explained why there is no projective plane of order 6.  He related the existence of a finite projective plane of order ''n'' to the existence of a hyper-Graeco-Latin square."</ref> 
*7 : all isomorphic to PG(2, 7), configuration (57<sub>8</sub>) with 5,630,688 automorphisms
*8 : all isomorphic to PG(2, 8), configuration (73<sub>9</sub>) with 98,896,896 automorphisms, 6 higher by prime power, 8=2<sup>3</sup>
*9 : PG(2, 9), and three more different (non-isomorphic) [[Non-Desarguesian plane#Examples|non-Desarguesian planes]]: a [[Hughes plane]], a [[Hall plane]], and the dual of this Hall plane. All are described in {{harv | Room | Kirkpatrick | 1971}}.
*10 : impossible as an order of a projective plane, proved by heavy computer calculation.{{sfnp|Lam|1991}}
*11 : at least PG(2, 11), others are not known but possible.
*12 : it is conjectured to be impossible as an order of a projective plane.{{cn|date=November 2024}}