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

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

Thenon-solvable triangle transitive planes

The class of finite translation planes admitting non-solvable triangle transitive groups is completely determined as the class of irregular nearfield planes admitting non-solvable groups

The classification of inherited hyperconics in Hall planes of even order

AbstractIn this note we complete the classification of inherited hyperconics in Hall planes of even order that was started by O’Keefe and Pascasio by proving that in the cases left open in [C.M. O’Keefe, A.A. Pascasio, Images of conics under derivation, Discrete Math. 151 (1996) 189–199] there are no inherited hyperconics

Blocking and double blocking sets in finite planes

In this paper, by using properties of Baer subplanes, we describe the construction of a minimal blocking set in the Hall plane of order q(2) of size q(2) + 2q + 2 admitting 1-,2-,3-,4-, (q + 1)- and (q + 2)-secants. As a corollary, we obtain the existence of a minimal blocking set of a non-Desarguesian affine plane of order q(2) of size at most 4q(2)/3 + 5q/3, which is considerably smaller than 2q(2) - 1, the Jamison bound for the size of a minimal blocking set in an affine Desarguesian plane of order q(2). We also consider particular Andre planes of order q, where q is a power of the prime p, and give a construction of a small minimal blocking set which admits a secant line not meeting the blocking set in 1 mod p points. Furthermore, we elaborate on the connection of this problem with the study of value sets of certain polynomials and with the construction of small double blocking sets in Desarguesian projective planes; in both topics we provide some new results