New Results In Finite Geometries Pertaining To Albert-like Semifields

Abstract

One of the most widely studied class of semifields are the generalized twisted fields defined by Albert in the 50s and 60s. The collineation groups of generalized twisted field planes have been completely described. In a series of papers Cordero and Figueroa studied semifields with an autotopism that acts transitively on one side of the autotopism triangle, equivalently the plane admits an autotopism which induces a permutation on a side of the autotopism triangle of order a p-primitive divisor of pr - 1. They showed that with some minor exceptions the plane is a generalized twisted field plane. These planes are coordinatized by pre-semifields (K,+, o) where x o y = (sigma) n-1 i=0 aix(i)y(ei) for x, y K = GF(pn). Hence either (pi) is a generalized twisted field plane or there exist at least two non-zero indices u; v such that au 6= 0 6= av. In this case the pre-semifield has the product x o y = xy+aux(u)y(eu)+avx(v)y(ev). In this work we study in depth the case in which there are precisely two nonzero indices. In this case the multiplication behaves much like a generalization of Albert's generalized twisted fields. For many of the cases, these semifields are generalized twisted fields. We provide a variety of examples in which these semifields are not generalized twisted fields. For these we study the collineations of the semifield planes they coordinatize to help shed some light into the classification of the semifield planes of this type.