The following pages are but a brief summary of five lectures held at the conference “Groups -St Andrews 1985”. We recall basic definitions, state the main results and sketch some applications. Further motivation, detailed proofs and more examples will be published elsewhere. Some extra information can already be found in.

DEFINITIONS. PRELIMINARIES

Amalgamated sums

Let be a set of groups and Ф a set of homomorphisms Ф: AФ → BФ, with AФ, BФ ∈. The inductive limit of the system (,Ф), also called the amalgamated sum of relative to Ф, is a group X endowed with a set of homomorphisms ψA: A → x for all A ∈, such that, for all Ф ∈ Ф, AФ = BФ ˚ Ф the system (X, (ψA)) being universal for that property, in the usual sense. This characterizes (X,(ψA)) up to unique isomorphism. One can also view X as the group defined by the following presentation: the set of generators is the disjoint union of the elements of and the relations are those provided by the multiplication tab les of the elements of X plus all relations of the form Ф(a) = a for Ф ∈ Φ and a ∈ AФ.