Now we come to the tool that was the main subject of my thesis, and which, as far as I know, I am the first person to use in studying the problem in the context of groups (it has been used extensively in semigroups, and it has also been studied in rings and categories). This tool is dominions. Given a group G, a subgroup H of G, and a class of groups C for context, the dominion of H in G (in the class C) is the collection of all elements of G dominated by H (with respect to the class C); i.e. the interesection of all equalizers of pairs of maps f,g:G->K, with K in C, which contain H. The definition is due to John Isbell.

   The dominion construction has a number of very useful general properties: The dominion of H is always a subgroup of G, and contains H; if H is contained in H', then the dominion of H is contained in the dominion of H'; and the dominion of the dominion of H is always equal to the dominion of H. This means that we can view the dominion construction as a closure operator on the lattice of subgroups of G.

   It is not hard to show that the centralizer of the dominion of H equals the centralizer of H. In particular, the dominion of an abelian subgroup is always abelian. Also, the normalizer of the dominion of H contains the normalizer of H. Other general properties can also be established.

   In addition, we can derive certain other properties of the construction from general properties of the class of context C. For example: if the class C is closed under quotients, then a normal subgroup always equals its own dominion (simply compare the zero map z:G->G/N with the canonical map), and so in particular the dominion of H is always contained in the normal closure of H. If the class is closed under finite direct products, then the dominion of a product is the product of dominions. If the class is closed under taking subgroups, quotients, and squares (i.e., given a group G in C, it also contains GxG), then the dominion construction respects quotients.

   The dominion construction also allows me to place my work in the context of Universal Algebra and similar work done with semigroups.

   But how is the dominion construction connected to amalgams?

   Say we have a special amalgam of G over H, in the class of context C. Say we have a candidate for embeddability M, together with two embeddings i,j:G->M. Since the core of the amalgam is H, then necessarily i and j agree on H. But then, by construction, they also agree on the dominion of H in G. Thus, the amalgam cannot be strongly embedded unless H equals its own dominion. Under mild closure conditions on C, the converse is also true. In fact, the dominion of H equals the smallest subgroup D of G which contains H and such that the special amalgam of G over D is strongly embeddable in C.

   In general, we might say that dominions measure how hard it is to "upgrade" a weak embedding into a strong embedding. That is, if we have an amalgam which we know is weakly embeddable, and want to know whether it is also strongly embeddebable. Intuitively, we have:

   The larger the dominion of a subgroup tends to be with respect to the subgroup in a given class, the harder it is to upgrade a weak embedding into a strong embedding.

    These are, of course, general statemnts; if an 'arbitrary' dominion is likely to be very large, then an arbitrary amalgam which is weakly embeddable is unlikely to be strongly embeddable. In one extreme, we do have a strong result:

   Theorem. Suppose that a class C satisfies certain mild technical conditions. If all dominions are trivial (that is, every subgroup equals its dominion) in the class C, then any weakly embeddable amalgam is also strongly embeddable.

   In general, when dealing with dominions, what we want is a description of the dominion in terms of G and H (and, of course, C, which is usually implicit). There are two kinds of subgroups in which we are particularly interested. Thinking of the dominion construction as a closure operator, we are certainly interested in those subgroups which are "closed." These correspond to subgroups equal to their own dominion, and hence subgroups over which the corresponding special amalgam is strongly embeddable.

   In analogy to the compact topological spaces, we are also interested in the groups H which are not only closed in a given G, but which are closed in any overgroup G in the class. These groups are called absolutely closed, and a little thought will show that they are nothing more than the special amalgamation bses for the class.

   The second class of subgrups we are particularly interested in (and the class Isbell was thinking of when he defined dominions) are those proper subgroups which are dense in G, i.e. for which the dominion equals all the group G. These are related to epimorphisms, which is the subject of the next section.