Part I received February 13, 1947. The greater part of the material in this paper was presented as a D.Phil. thesis at the University of Oxford.
Part II received August 27, 1947.
PART I. Definitions and General Properties
In his paper "Die Untergruppen der freien Produkte" A. Kurosch determines completely the structure of the subgroups of a free product of groups. He also suggests the problem of determining similarly the subgroups of a free product of groups with an amalgamated subgroup. This problem will be solved completely. However, to describe the structure of these subgroups adequately, a generalization of the free product with one amalgamated subgroup is necessary. It is with this generalization that the first part of our investigations is mainly concerned; the problem of determining the subgroups of free products with one amalgamated subgroup and of generalized free products with amalgamated subgroups will be dealt with in the second part.
The concepts introduced so far may be looked upon as a special case of a rather more general problem. In 4 we introduce the notion of an "incomplete group." Two questions arise: whether an incomplete group is imbeddable into a group; and if it is imbeddable, whether there exists a "largest" group which contains the incomplete group and is generated by it. This second question will be answered in the affirmative. That the answer to the first question is not always positive, is known. Since the system of groups Gα with amalgamated subgroups Uαβ must form an incomplete group, if the generalized free product is to exist, the example in 3 provides an instance of a different type of incomplete group which is not imbeddable.
In 5 we derive a necessary and sufficient criterion for the existence of the generalized free product with amalgamated subgroups reducing the general case to a more special (though by no means easier) case. In 6 we describe the structure of the generalized free product in a somewhat different way. This leads to a number of special results and examples (7,8). These enable us to derive some simple and useful sufficient criteria for the existence of the generalized free product in the case of three factors. however, they also seem to indicate that more general results in this direction can hardly be hoped for (9). We conclude the paper with a detailed discussion of the free product of three infinite cycles with amalgamated subcycles.
PART II. The Subgroups of Generalized Free Products
In Part I we defined the generalized free product G of the the groups Gα with amalgamated subgroups Uαβ. We found that G contains a group U which is itself a free product with amalgamated subgroups Uαβ. Its factors are the subgroups Uα of Gα which are generated by all the amalgamated subgroups Uαβ of Gα. If, in particular, all Uαβ are identical, then Uα = Uαβ for all β, all the groups Uα are isomorphic, and G is simply the free product of all the groups Gα with the one amalgamated subgroup U. In the general case the subgroup U of G may be loosely described as the smallest generalized free product with the amalgamated subgroups Uαβ.
We are going to show that every subgroup H of G is itself a generalized free product—unless H belongs to U, in which case nothing can be said.