The abelian Singer-group problem for finite projective planes and the unfinished case (3) of the classification of finite hyperfield extensions of the Krasner hyperfield

A finite projective plane is said to admit a Singer group when a subgroup of its collineation group acts sharply transitively (simply transitively) on the point set. The classical conjecture, going back to a question of Hall from 1947 and strongly supported by results of Wagner and Ostrom, asserts that every finite projective plane with a cyclic Singer group is Desarguesian, i.e. the classical plane over a finite field. Connes and Consani showed that this question embeds naturally in hyperring theory: their classification of finite commutative hyperfield extensions of the Krasner hyperfield $\mathbb{K}$ has three cases — group hyperfields $\mathbb{K}[G]$ built from finite abelian groups, quotients $F_{q^m}/F_q^{\times}$ of finite fields, whose associated geometry is Desarguesian, and one remaining case, in which the hyperfield's multiplicative group is the incidence group of a finite non-Desarguesian projective plane with a simply transitive abelian group of collineations. Cases (1) and (2) are completely classified; case (3) is open and is equivalent to the existence of a non-Desarguesian finite projective plane admitting an abelian Singer group. The open case is: does a finite hyperfield extension of the Krasner hyperfield of type (3) exist — equivalently, is there a non-Desarguesian finite projective plane with an abelian collineation group acting sharply transitively on its points? No such hyperfield is known to exist, and no proof of non-existence is known.

Background

A projective plane is an incidence structure of points and lines in which any two points lie on a unique line, any two lines meet in a unique point, and there exist four points no three of which are collinear. A finite projective plane of order $n$ has $n^2+n+1$ points and equally many lines. The Desarguesian plane $\mathrm{PG}(2,\mathbb{F}_q)$ over the finite field $\mathbb{F}_q$ is the standard model; a plane is called Desarguesian when it is isomorphic to such a plane (equivalently, when it is coordinatized by a field, or satisfies the classical Desargues configuration), and a long-standing theme of finite geometry is to understand how far non-Desarguesian (i.e. non-classical) finite planes can deviate from it. A collineation of a plane is an incidence-preserving permutation of its points and lines; a group of collineations is sharply transitive (or simply transitive, or regular) on points when it maps any point to any other point in exactly one way. A collineation group acting sharply transitively on the points of a finite plane is called a Singer group of the plane, and a cyclic one a Singer cycle, since Singer showed in 1938 that every Desarguesian finite plane carries such a cyclic group. Hall conjectured in 1947 that the converse holds for cyclic groups: a finite projective plane admitting a Singer cycle is Desarguesian. The conjecture is open; finiteness is essential, since Hall constructed infinite non-Desarguesian planes with infinite cyclic Singer groups. By a result of Karzel (1965), as recorded by Thas (2016), a finite abelian Singer group is automatically cyclic, so the conjecture is equivalent to the statement that every finite plane with an abelian Singer group is Desarguesian; equivalently, no non-Desarguesian finite projective plane admits an abelian collineation group acting sharply transitively on points. No example of such a plane is known, and, as recorded by Thas, even the possible prime-power order of the underlying plane is undetermined. The problem predates Connes and Consani; they connected it to hyperring theory. A hyperring is an algebraic structure like a ring but with a multivalued addition: the sum of two elements is a nonempty subset of the hyperring, with associativity, commutativity, and distributivity adapted accordingly. A hyperfield is a hyperring in which the nonzero elements form a group under multiplication and the additive structure satisfies field-like axioms. The Krasner hyperfield $\mathbb{K}$ is the hyperfield on the set $\{0,1\}$ with usual multiplication and multivalued addition given by $1+1=\{0,1\}$ and $1+0=1$. The hyperfield $\mathbb{K}$ encodes the crude additive structure of any field, and a morphism of hyperrings (in the sense of Connes and Consani, where the multivalued sum of images must be contained in the image of the sum) mediates between field arithmetic and sign-like or incidence-like combinatorics. Connes and Consani classified the finite commutative hyperfield extensions of $\mathbb{K}$ (Theorem 3.11): each such extension is either of type (1), the hyperfield $\mathbb{K}[G]$ attached to a finite abelian group $G$, whose associated projective geometry is a single line; of type (2), a quotient $F_{q^m}/F_q^{\times}$ of a finite field by its multiplicative subgroup $F_q^{\times}$, whose associated geometry is the Desarguesian space $\mathrm{PG}(m-1,q)$ — for $m=3$ a Desarguesian plane with its (cyclic) Singer group $F_{q^3}^{\times}/F_q^{\times}$; or of type (3), a hyperfield whose multiplicative group is the incidence group (i.e. the group of points of the plane, with left and right translations automorphisms of the incidence structure) of a finite non-Desarguesian projective plane with a simply transitive abelian group of collineations. Types (1) and (2) are fully determined; type (3) corresponds exactly to the hypothetical non-Desarguesian plane with an abelian Singer group, and no example is known. This bridge makes the classification a question of both finite geometry and modern algebra.

Problem Statement

Does there exist a non-Desarguesian finite projective plane admitting an abelian collineation group acting sharply transitively (i.e. simply transitively) on its points? Equivalently, in the language of hyperfields introduced by Connes and Consani, does case (3) of their classification of finite commutative hyperfield extensions of the Krasner hyperfield $\mathbb{K}$ contain a genuine example — namely, an extension whose multiplicative group is the incidence group of such a plane? A complete answer either constructs such a plane or hyperfield, or proves that none exists, thereby completing the classification of finite commutative hyperfield extensions of the Krasner hyperfield and establishing the abelian (equivalently cyclic) Singer-group conjecture for finite projective planes.

References

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