Structures of circular planar nearrings

Abstract

The family of planar nearrings enjoys quite a few geometric and combinatoric properties. Circular planar nearrings are members of this family which have the character of circles of the complex plane. On the other hand, they also have some properties which one may not find among the circles of the complex plane. In this dissertation, we first review the definition and characterization of a planar nearring, and some various ways of constructing planar nearrings, as well as various ways of constructing BIBD's from a planar nearring. Circularity of a planar nearring is then introduced, and examples of circularity planar nearrings are given. Then, some nonisomorphic BIBD's arising from the same additive group of a planar nearring are examined. To provide examples of nonabelian planar nearrings, the structures of Frobenius groups with kernel of order 64 are completely determined and described. On the other hand, examples of Ferrero pairs (N, Φ)'s with nonabelian Φ, which produce circular planar nearrings, are provided. Finally, we study the structures of circular planar nearrings generated from the finite prime fields from geometric and combinatoric points of view. This study is then carried back to the complex plane. In turn, it gives a good reason for calling a block from a circular planar nearring a "circle."