It is well known that in the category of all groups and homomorphisms, every epimorphism is onto. This result does not hold for certain other categories of groups. The condition that an epimorphism θ:G → A is not onto is equivalent to the condition that θ(G) is a proper subgroup A with the property that any two homomorphisms α, β on A which agree elementwise on θ(G) must agree on A. Such a subgroup can be called dense (see e.g. [1]). Naturally the existence of such homomorphisms α and β depend on the particular class of groups that is available. We will choose to work within the context of varieties even though many of the results will hold true for more modest classes of groups.