The existence of a projective plane of order 12

The existence of a finite projective plane of order 12 — equivalently, of a symmetric $2$-$(157,13,1)$ design, or of a complete set of 11 mutually orthogonal Latin squares of order 12 — is the smallest open case of the conjecture that finite projective planes exist only for prime-power orders. The Bruck–Ryser theorem is silent at order 12 ($12 \equiv 0 \pmod 4$), and the code-based computational strategy that settled order 10 in 1989 has no known extension to order 12. Akiyama and Suetake (2008) eliminated collineation groups of order 8, thereby confining the full collineation group of a putative plane of order 12 to orders dividing 4 or 9. Follow-up work by the same group eliminated order 9 (2019) and order 4 (2023), so any plane of order 12 must now have collineation group of order 1, 2, or 3 — the symmetry-reduction route is nearly exhausted while the central question stands.

Background

A finite projective plane of order $n$ is an incidence structure of $n^2+n+1$ points and the same number of lines in which every line is incident with $n+1$ points, every point lies on $n+1$ lines, any two distinct points lie on exactly one common line, and any two distinct lines meet in exactly one point. Equivalently, it is a symmetric $2$-$(n^2+n+1, n+1, 1)$ design: a block design with equally many blocks and points, blocks of size $n+1$, in which every pair of distinct points is contained in exactly one block. For every prime power $q$ there is a projective plane of order $q$, the desarguesian plane $PG(2,q)$ coordinatized by the finite field with $q$ elements, and every projective plane discovered to date has prime-power order. The standing conjecture is that this is necessary: no finite projective plane of non-prime-power order exists.

Nonexistence is known only in fragments. The Bruck–Ryser theorem (1949) states that if $n \equiv 1$ or $2 \pmod 4$, then a plane of order $n$ can exist only if $n$ is a sum of two integer squares; among orders up to 25 this excludes $n = 6, 14, 21, 22$, but it says nothing when $n \equiv 0$ or $3 \pmod 4$ — in particular nothing about $n = 10$ or $n = 12$. Order 10 was settled in 1989 by Lam, Thiel and Swiercz: an exhaustive computer search (about 2000 hours on a Cray supercomputer, working through the possible weight enumerators of the binary code spanned by the rows of a putative incidence matrix) proved that no plane of order 10 exists. The next composite order not covered by Bruck–Ryser, namely 12, is therefore the smallest order for which the existence of a projective plane is undecided. The order-12 question has two standard equivalent forms: it asks whether a symmetric $2$-$(157,13,1)$ design exists, and whether a complete set of $n-1 = 11$ mutually orthogonal Latin squares of order 12 exists (Latin squares of the same order no two of which share an entry in any cell position; a complete set of $n-1$ of them is equivalent to a plane of order $n$).

Because no direct attack has succeeded, work on order 12 has proceeded by eliminating symmetry. A collineation of a plane is an automorphism of the incidence structure — a permutation of the points together with a permutation of the lines that preserves incidence — and the collineation group is the group of all collineations. A nonidentity collineation of a finite plane is either planar, meaning its fixed points and fixed lines themselves form a subplane, or a generalized elation or homology, meaning it fixes a line (the axis) pointwise and a point (the center) linewise. Janko and van Trung began the modern study of order 12 in 1980 and, in a series of papers through 1984 (Janko as sole author of the last), proved that the full collineation group of any projective plane of order 12 is a $\{2,3\}$-group (its order is divisible by no prime other than 2 and 3), that a collineation group of order 4 must be cyclic (in particular no four-group $V_4$ acts on the plane), that a group of order 6 must be abelian, that no elation of order 3 exists, and that no collineation group of order 27 exists.

Horvatić-Baldasar, Kramer and Matulić-Bedenić (1986–1987) sharpened the structural restriction: the full collineation group of a plane of order 12 is either a 2-group of order at most 16 or a 3-group of order at most 9 — in the shorthand used in this literature, its order divides 16 or 9. Both groups of order 6 (abelian and non-abelian) were excluded (Horvatić-Baldasar–Kramer–Matulić-Bedenić 1986; Janko–van Trung 1981). Suetake (2004) then excluded collineation groups of order 16, by a computer-assisted proof that certain orthogonal arrays $OA(72,12,6,2)$ do not exist. Akiyama and Suetake (2008) excluded groups of order 8: if a plane of order 12 admitted a collineation group of order 8, contracting the plane through the involutory elation in that group would produce a symmetric transversal design $\mathrm{STD}_2[12;6]$ — an incidence structure with 72 points and 72 blocks of size $k = \lambda u = 12$, whose points and blocks partition into classes of size $u = 6$ so that two points from different classes lie on exactly $\lambda = 2$ common blocks — carrying a semiregular automorphism group of order 4 (one whose nonidentity elements fix neither a point nor a block), and an orbit-matrix enumeration shows that no such design exists. As its terminal corollary, the paper confined the full collineation group of a putative plane of order 12 to orders dividing 4 or 9, with any group of order 4 cyclic — while the existence question itself remained open, and remains open today.

Problem Statement

Does there exist a projective plane of order 12?

Equivalent standard forms of the same question: does there exist a symmetric $2$-$(157,13,1)$ design, that is, an incidence structure with 157 points and 157 blocks in which every block is incident with exactly 13 points, every point lies on exactly 13 blocks, and every pair of distinct points lies on exactly one common block? Equivalently, does there exist a complete set of 11 mutually orthogonal Latin squares of order 12?

Any projective plane of order 12 would have full collineation group of order 1, 2, or 3, and no known result excludes any of these three cases.

References

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