Editing Finite geometry (section)

=== Finite projective planes ===

A projective plane geometry is a nonempty set ''X'' (whose elements are called "points"), along with a nonempty collection ''L'' of subsets of ''X'' (whose elements are called "lines"), such that:
# For every two distinct points, there is exactly one line that contains both points.
# The intersection of any two distinct lines contains exactly one point.
# There exists a set of four points, no three of which belong to the same line.
[[File:Fano plane Hasse diagram.svg|thumb|200px|left|Duality in the [[Fano plane]]: Each point corresponds to a line and vice versa.]]
An examination of the first two axioms shows that they are nearly identical, except that the roles of points and lines have been interchanged.
This suggests the principle of [[Duality (mathematics)#Dimension-reversing dualities|duality]] for projective plane geometries, meaning that any true statement valid in all these geometries remains true if we exchange points for lines and lines for points.
The smallest geometry satisfying all three axioms contains seven points. In this simplest of the projective planes, there are also seven lines; each point is on three lines, and each line contains three points.
[[Image:Fano plane.svg|thumb|right|The [[Fano plane]] ]]

This particular projective plane is sometimes called the '''[[Fano plane]]'''.
If any of the lines is removed from the plane, along with the points on that line, the resulting geometry is the affine plane of order 2.
The Fano plane is called '''the''' ''projective plane of order'' 2 because it is unique (up to isomorphism).
In general, the projective plane of order ''n'' has ''n''<sup>2</sup>&nbsp;+&nbsp;''n''&nbsp;+&nbsp;1 points and the same number of lines; each line contains ''n''&nbsp;+&nbsp;1 points, and each point is on ''n''&nbsp;+&nbsp;1 lines.

A permutation of the Fano plane's seven points that carries [[incidence (geometry)|collinear]] points (points on the same line) to collinear points is called a [[collineation]] of the plane. The full [[collineation group]] is of order 168 and is isomorphic to the group  [[PSL(2,7)]] ≈ PSL(3,2), which in this special case is also isomorphic to the [[general linear group]] {{nowrap|GL(3,2) ≈ PGL(3,2)}}.