A Theoretical and Computational Investigation of AG-groups

Abstract

An AG-group (also called LA-group) is an AG-groupoid having left identity and inverses. AG-groups have been originally introduced by M.S. Kamran. This thesis concentrates on the study of AG-groups as a special class of quasigroups and as a general class of abelian groups both from a theoretical and computational point of view. We investigate several properties of abelian groups which can be generalized to the non-associative structure of AG-groups. Similarly several properties of loops are shown to hold in AG-groups. We introduce Bol∗ quasigroups, a generalization of AG-groups, as well as some interesting subclasses of AG-groups. We present both a computational and an algebraic enumeration of AG-groups. Computationally we can enumerate them up to order 11 but algebraically we are able to count them of any given order. A further motivation to study AG-groups is that they have two-sided unique inverses and can therefore be seen as a generalization of two-sided loops. Consequently we provide investigation of two such loops (i) C-loops and (ii) Jordan loops. This leads to the enumeration of Jordan loops up to order 10 which is also a part of this thesis. Furthermore, we investigate the application of AGgroups in geometry and exploit AG-groups to map out the general structure of AG-groupoids and further explore cancellativity of AG-groupoids. In spite of a great deal of investigations of AG-groupoids and their subclasses for nearly four decades no progress had been made in obtaining enumeration results. In fact, not even the exact number of non-associative AG-groupoids for the order 3, the smallest possible order, was known up to now. We are enumerating AG-groupoids up to order 6 for the first time. We then classify our data and give twenty four new classes of AG-groupoids. The relations between them to each other have been established. In the end we have collected all our findings into a GAP package ”AGGROUPOIDS” that contain several functions that check various characteristics of a given Cayley table of AG-groupoids and AG-groups.