Absolute convexity in ordered groups.

The object of this thesis is to study the problem posed by L. Fuchs concerning the possible existence of subgroups of an ordered group which are convex in every order on the group; such subgroups are called absolutely convex subgroups. The first two sections of this thesis contain definitions and general results from standard text books on the theory of groups with an emphasis on the theory of ordered groups. In Section II we also establish some general results on ordered groups. Then the concept of absolute convexity is introduced and later some straightforward properties of absolutely convex subgroups are proved. Section III is devoted to the study of absolute convexity in nilpotent groups. This section contains the main result of this thesis in which a necessary and sufficient condition for a subgroup to be absolutely convex in a nilpotent group is established. This theorem is then applied to determine various results about absolutely convex subgroups of nilpotent groups. These results combined with the theory of basic commutators are used in Section IV to consider the situation in free nilpotent groups and it has been possible to solve completely the problems arising in this case. We also give some examples of absolutely convex subgroups, end then use absolute convexity to construct all the orders on a torsion free nilpotent group on 2 generators, when the class is not more than 5. Towards the end of this section an example of a nilpotent group of class n is constructed in which every member of the upper central series is absolutely convex. Finally Section V consists of the embedding of a torsion free abelian group A of tank not more than 7 as the absolutely convex centre of a torsion free nilpotent group of class 2. It is further conjectured that such an embedding is possible when the rank of A is arbitrary.