Projective Plane of Non Prime-Power Order

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Does there exist a projective plane of non prime-power order?

Simply: On an ordinary plane, two nonparallel lines intersect in a single point.

The concept of a projective plane allows every two lines to intersect (for example, by adding a common point at infinity). The definition of a finite projective plane requires any two lines to intersect in exactly one point, but does not require the lines to be ‘straight’.

In detail: A finite projective plane is a set of points, say P = {p1,p2,..,pn}, and a set of lines, say L = [l1,l2,...,ln], where (a) every line contains the same number of points; (b) every point lies on the same number

of lines; (c) every two points lie together on exactly one line; and (d) every two lines intersect in exactly one point.

Suppose each line contains k+1 points. An easy counting argument shows that each point lies on k+1 lines, and the properties (c) and (d) imply that the number of points and lines is k2 + k + 1. In this case, we call the pair (P,L) a projective plane of order k.

The Fano Plane, left, is a projective plane of order 2, the smallest such plane, with only seven points and seven lines.

The problem is finding the possibilities for k. It is relatively easy to show that there exists a projective plane of order k whenever k is a prime number or a prime multiplied by itself a number of times, such as 4 (= 22) or 8 (= 23) or 1331 (= 113). It is not known if there are projective planes of order 12, 15, 18, 20, and so on.

Discipline: Linear algebra, projective geometry, combinatorics.

Progress: In 1901 Gaston Tarry proved that there is no projective plane of order 6, in answering Euler’s ‘Thirty-six officers problem’ of 1782. The 1949 Bruck-Ryser theorem implies that there is no projective plane of order k if k–1 or k–2 is divisible by 4 and k is not the sum of two squares. In 1991, Clement Lam eliminated the possibility of a projective plane of order 10 with the help of a massive computer calculation.

NZIMA connection: This question was discussed at conferences in the NZIMA’s programmes on Combinatorics and Geometry: Interactions with Algebra and Analysis.

Published in IMAges 11 - October 2011