On the Multiplicative Structure of Quasifields and Semifields: Cyclic and Acyclic Loops

Abstract

This note is concerned with the multiplicative loop  $L$ of a finite quasifield or semifield, and the associated geometry. It investigates when the principal powers of some element of the multiplicative loop $L$ ranges over the  whole loop: in this situation the loop $L$ is cyclic (or primitive) and is acyclic otherwise. A conjecture of Wene essentially asserts that a finite semifield cannot be acyclic.No counterexamples to the Wene conjecture are known for semifields of order $>32$; in fact, in many situations the Wene conjecture is known to hold, as established in various papersby Wene, Rùa and Hamilton. The primary aim of this note is to show that, in contrast to the above situation,there exists at least one acyclic quasifield for every square prime powerorder $p^{2r}>4$. Additionally, we include a simple conceptual proof ofa theorem of  Rùa, that establishes the primitivity ofthree-dimensional semifields.