Again, we have a group G, a subgroup H, and a class C of groups for context, and we ask which elements of G are dominated by H with respect to C.

   The case where the dominion equals all of G is of particular interest. In this case, we say that H is dense in G; the term comes from topology, if we consider the dominion construction as a closure operator. It means that a morphism in C with domain G is completely determined by its value on H.

   This means that any pair of maps f,g:G->K, with K in C, which agree on H are actually equal. Another way to state this is by letting i:H->G be the inclusion map, and saying that if fi=gi, then f=g. In other words, the map i is right-cancellable in C.

   Right-cancellable morphisms have a special name in Category Theory: they are called epimorphisms; they are the dual concept of the monomorphisms, or left-cancellable morphisms. In most of Algebra, monomorphism is simply another word for injective map, since it can be shown that the injective maps and the monomorphisms are exactly the same.

   It is also easy to verify that a surjective map on underlying sets is necessarily an epimorphism. The converse, however, is not true in general. Although it is true in the category of all Groups, it is false in other common categories of algebras, such as the category of semigroups (the embedding on the natural numbers into the integers is a non-surjective epimorphism), and the category of rings (the embedding of the integers into the rationals is a nonsurjective epimorphism). One wants to obtain a description of which maps are epimorphisms, whenever possible; for example, in the category of Hausdorff spaces, a morphism is an epimorphism if and only if the image is dense (in the usual topological sense).

   Even if we restrict our attention to groups, there are natural classes of groups where not all epimorphisms are surjective. B.H. Neumann proved, for example, that the embedding of the alternating group on four letters into the alternating group on five letters is a nonsurjective epimorphism in the variety generated by the latter. I have constructed other examples, which I will discuss more fully below.

   The concept of dominion was in fact introduced by Isbell because he was interested in studying epimorphisms, and particularly the nonsurjective ones. There are some who argue that studying and classifying epimorphisms is one of the main objectives of Algebra.

   Although I don't go that far, I do believe that classifying epimorphisms, and particularly nonsurjective epimorphisms, is pretty important. If nothing else, they tell us that the behavior of a group under morphisms is completely determined by a proper subgroup, which implies a rather rigid structure. And also, there are a number of categorical constructions and concepts which are restricted to epimorphisms, and although they may be uninteresting when the epimorphism is surjective, they become rather more important when there are nonsurjective ones (an example of this is the notion of a 'projective').