Techniques for Enumerating Veblen-Wedderburn Systems

Veblen-Wedderburn systems (defined in the next section) are algebraic sy terns tha t may be used to coordinatize affine planes and thereby proiecti~ planes. Such planes may be characterized by the fact tha t they satisfy a certa geometric configuration, Hall's Theorem L [2]. Conversely, all planes satisfyii Theorem L, which are not Desarguesian, give rise to many non-isomorph Veblen-Wedderburn systems. The reader who is unfamiliar with these resul will find a very readable account of them in the Slaught Memorial Papers N 4, "Contributions to Geometry" (Am. Math. Month. 62 (1955), pt. II) in paper by R. H. Bruck entitled "Recent Advances in Euclidean Plane Geometry One of the problems is to determine all Veblen-Wedderburn planes of a giw order. I t turns out that the best way of handling this is to determine all Veble Wedderburn systems first. The case of 16 elements is the smallest one requiril the aid of a computer. Considerable mathematical theory [Theorems 1 and below] is required, however, before the computer is called into play. A cor plete account of the theory is presented here. However, the task of sorting tl computed Veblen-Wedderburn systems into isomorphic classes and then arran ing them into geometries is too lengthy to be included here. Only the final resul are tabulated. The nucleus of a Veblen-Wedderburn system with 16 elements may be GF(~ GF(4) or GF(16). When the nucleus is GF(16), the plane is Desarguesian ai no other Veblen-Wedder~mrn system gives rise to the plane besides GF(16) itse When the nucleus is GF(4) we determine all possible non-isomorphic Veble Wedderburn systems. I t turns out tha t there are 75 of them and that th~ determine two distinct projective planes P(1) and P(2). P(1) is determined 25 of the Veblen-Wedderburn systems, which include the known Hall syster [2, p. 274]. P(2) is a new plane and in fact five of its 50 Veblen-Wedderbu systems are division rings, representing the first known division rings with elements. The computations were carried out on SwAc. 1 Eventually, a check on tl computations of SwAc was obtained through geometric considerations. Neve theless, invaluable time was saved as a result of the computations. In the case where the nucleus is GF(2) a systematic enumeration of all tl