Our odyssey begins with nilpotent groups, which are in many ways the second simplest kind of groups there are (the simplest being abelian groups). An abelian group is defined by the rule that the commutator subgroup [G,G] of G must be trivial. We write our commutators left normed; i.e. [x,y,z]=[[x,y],z], where

   We define the lower central series of G by setting G(0)=G, and setting G(n+1)=[G(n),G]. If for some n we have that G(n)={e}, then G is nilpotent. The nilpotency class of G is, in that case, the smallest n for which G(n) is trivial. We then say that G is nilpotent of class n. We say that a group G is m-nil if it is of nilpotency class at most m.

   Note for example that a group is abelian if and only if it is nilpotent of class 1; and that a group is 2-nil if and only if the commutator subgroup is central.

   The class of all m-nil groups is a variety, and we denote it by Nm. For m=1 we usually simply write A, for abelian groups. The class of all nilpotent groups (of any class) is not, however, a variety, since it is not closed under arbitrary direct products; it is only a quasivariety.

   When studying dominions, varieties of abelian groups are uninteresting, since all subgroups are normal and hence all dominions trivial. So it makes sense to start our study with the nilpotent groups. In fact, we shall begin with the nilpotent groups of class two.