Infinite Simple Bol Loops

If the left multiplication group of a loop is simple, then the loop is simple. We use this observation to give examples of infinite simple Bol loops.Comment: 4 pages, AMS-LaTeX, to appear in Comment. Math. Univ. Carolinae for a special issue: the Proceedings of Loops03. Version 3: more minor changes suggested by the refere

Overlaps in Field Generated Circular Planar Nearrings

We investigate circular planar nearrings constructed from finite fields as well the complex number field using a multiplicative subgroup of order $k$, and characterize the overlaps of the basic graphs which arise in the associated $2$-designs

Theory of K-loops

The book contains the first systematic exposition of the current known theory of K-loops, as well as some new material. In particular, big classes of examples are constructed. The theory for sharply 2-transitive groups is generalized to the theory of Frobenius groups with many involutions. A detailed discussion of the relativistic velocity addition based on the author's construction of K-loops from classical groups is also included. The first chapters of the book can be used as a text, the later chapters are research notes, and only partially suitable for the classroom. The style is concise, but complete proofs are given. The prerequisites are a basic knowledge of algebra such as groups, fields, and vector spaces with forms

Theory of K-loops

x, 186 p. ; 24 c

Relatives of K-loops: Theory and examples

summary:A {\it K-loop\/} or {\it Bruck loop\/} is a Bol loop with the automorphic inverse property. An overview of the most important theorems on K-loops and some of their relatives, especially Kikkawa loops, is given. First, left power alternative loops are discussed, then Kikkawa loops are considered. In particular, their nuclei are determined. Then the attention is paid to general K-loops and some special classes of K-loops such as 2-divisible ones. To construct examples, the method of {\it derivation\/} is introduced. This has been used in the past to construct quasifields from fields. Many known methods to constructing loops can be seen as special cases of derivations. The examples given show the independence of various axioms

On the extension of involutorial Bol loops

The group theoretical problem of the existence of system of representatives T of the subgroup H of G such that T consists of conjugacy classes of involutions leads to the theory of Bol loops of exponent 2. In this paper, we develop a theory of extensions of such loops and give two applications of the theory. First, we classify all Bol loops of exponent 2 of order 16; second, we classify all (left) Bol loops of exponent 2 whose right nucleus has index 2. In particular, we give a class of examples of non-nilpotent such Bol loops