We come now to a secondary but nonetheless important question. Which subclasses of groups are "reasonable" classes to which we should restrict our attention for our guiding question? The answer will depend on your background or the applications or tools you have in mind.

   Because of the tool I have chosen, dominions, the type of classes I am most interested in are the varieties of groups.

   A variety of groups is a class of groups which is closed under taking subgroups, arbitrary direct products, and homomorphic images (i.e. quotients). Equivalently, a variety of groups is the class of all groups that satisfy a given set of identities (for instance, the variety of abelian groups is defined by the identity xy=yx). They are the main kind of subclass in which Universal Algebra is interested, since they inherit many useful constructions: they have both products and coproducts, pushouts and pullbacks, and relatively free groups.

   Relatively free groups are particularly useful. They have the universal property of the free group: the relatively free group for a variety V on n generators is the unique group F(n), together with a set of generators x1, x2, ..., xn such that given any group G in V, and any set of n elements g1,...,gn of G, there exists a unique morphism f:F(n)->G such that f(xi)=gi.

   An excellent reference for varieties of groups is Hanna Neumann's book, Varieties of Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, New Series, Vol. 37, Springer Verlag.

   Another way to describe a variety is that they are all the groups which satisfy a given set of laws or identity. A law is simply a finite word on a finite number of variables; an identity is an equation between two words; we say that a group satisfies that identity if and only if every possible substitution of the variables by elements of the group yields an equality; and a group satisfies a law if and only if every substitution yields the identity elements. Thus, for example, the variety of abelian groups is defined by the law xy=yx.

   There are many examples of varieties which group theorists are used to working with: abelian groups, groups of a given exponent, nilpotent groups of a given class, solvable groups of a given class, etc. Or we may start with a collection of groups and work in the variety they generate: i.e. the smallest variety which contains them all. By the famous HSP Theorem of Birkhoff, this is the collection of all homomorphic images of all subgroups of all products of the groups we start with.

   There are other classes which look almost as good as varieties and which are also very common: for example, the class of all solvable groups, or the class of all nilpotent groups. Note that neither of those two classes are varieties, since they are not closed under arbitrary direct products. They are pseudovarieties; i.e. classes closed under subgroups, quotients, and finite direct products. If instead we allow infinite direct products but disallow quotients, we have quasivarieties.

   Almost all of my work has taken place in either varieties or pseudovarieties. Other classes may also be considered, however, and have been on occassion,