Editing Projective plane (section)

===Finite field planes===

By [[Wedderburn's little theorem|Wedderburn's Theorem]], a finite division ring must be commutative and so be a field. Thus, the finite examples of this construction are known as "field planes". Taking ''K'' to be the [[finite field]] of {{nowrap|1=''q'' = ''p''<sup>''n''</sup>}} elements with prime ''p'' produces a projective plane of {{nowrap|''q''<sup>2</sup> + ''q'' + 1}} points. The field planes are usually denoted by PG(2,&nbsp;''q'') where PG stands for projective geometry, the "2" is the dimension and ''q'' is called the '''order''' of the plane (it is one less than the number of points on any line). The Fano plane, discussed below, is denoted by PG(2,&nbsp;2). The [[#A finite example|third example above]] is the projective plane PG(2,&nbsp;3).

[[File:fano_plane_with_colored_lines.svg|thumb|The Fano plane. Points are shown as dots; lines are shown as lines or circles.]]

The [[Fano plane]] is the projective plane arising from the field of two elements. It is the smallest projective plane, with only seven points and seven lines. In the figure at right, the seven points are shown as small balls, and the seven lines are shown as six line segments and a circle. However, one could equivalently consider the balls to be the "lines" and the line segments and circle to be the "points" – this is an example of [[duality (projective geometry)|duality]] in the projective plane: if the lines and points are interchanged, the result is still a projective plane (see [[#Duality|below]]). A permutation of the seven points that carries [[incidence (geometry)|collinear]] points (points on the same line) to collinear points is called a ''[[collineation]]'' or ''[[symmetry]]'' of the plane. The collineations of a geometry form a [[Group (mathematics)|group]] under composition, and for the Fano plane this group ({{nowrap|1=PΓL(3, 2) = PGL(3, 2)}}) has 168 elements.