Michael J. Ganley

Commutative semifields, two dimensional over their middle nuclei

Then we say that S is a (finite) semzj?eld. The subset N, = (n E S: (xn)y = x(ny), Vx, y E S} is known as the middle nucleus of S. N, is a field, and S may be regarded as a (left or right) vector space over N,. A semifield in which multiplication is not associative (i.e., N, # S) is called proper. A finite semifield necessarily has prime power order and proper finite semifields exist for all orders q = p’, p prime, where Y > 3 if p is odd and r>4ifp=2. There is a considerable link between finite semifields and finite projective planes of Lenz-Barlotti class V.1. Indeed, any finite proper semilield gives rise to such a plane, and conversely any plane of the above type yields many isotopic, but not necessarily isomorphic, semifields. The paper by Knuth [4] is an excellent survey of finite semifields and their connections with projective planes. We mention one of the examples given in 141‘

Central Weak Nucleus Semifields

Using results of S. D. Cohen and the author, we characterize certain infinite families of finite semifields. This yields several infinite families of finite projective planes, all of which have order a power of 3, and which appear to be previously unknown.

Direct product difference sets

The theory of difference sets and relative difference sets has been used to investigate finite projective planes which admit a quasiregular collineation group of “reasonably” large order (see 14, 51.) In this paper we introduce the concept of a direct product difference set (and, in Section 6, a generalized d.p. difference set) to aid our investigation of quasiregular collineation groups. In Section 2 we define a d.p. difference set and show that such a difference set must have parameters of a particular form. We also give an explicit construction for an infinite family of these difference sets. In Sections 3 and 4 we consider a geometric object associated with the difference set and demonstrate the link with certain finite projective planes. In Section 5, this link is used to prove a result concerning the type of group which can contain a d.p. difference set. It is assumed, throughout, that the reader is familiar with finite projective planes (for instance, [l, 61) and difference sets (in particular, [3]).

The translation planes admitting a nonsolvable doubly transitive line-sized orbit

The finite translation planes of ordern not in {34,36,112,192,292,592 that admit a non-solvable doubly transitive line-sized orbit are completely classified.

On a conjecture of Kallaher and Liebler

One form of the conjecture referred to in the title states that for a finite non-Desarguesian semified plane, π, the number of isotopic, but non-isomorphic, semifields which can coordinatize π is at least 5. The case when the autotopism group of π is solvable was essentially dealt with by Kallaher and Liebler. Here, using results of Hering and Liebeck derived from the classification of the finite simple groups, we complete the proof of this conjecture.

Difference sets and planar polarities

A block design is a finite set p of elements called points , where | p | = v , together with certain distinguished subsets of p called blocks , such that (i) each block contains k points, (ii) each point is contained in r blocks, and (iii) two distinct points are contained in precisely λ blocks.