The next broad class of groups after abelian and nilpotent are the solvable groups. Essentially, they are groups which are "build up" from abelian groups. Explicitly, we define the derived series of G to be given by: G^(0)=G; G^(n+1)=[G^(n),G^(n)]. A group is solvable if there is some m such that G^(m)={e}, and the least such m is the length of G.

   Since we have that G^(n)/G^(n+1) is abelian, a solvable group is constructed through successive extensions by abelian groups. Note that a nilpotent group is necessarily solvable, but the converse does not hold.

   This is perhaps also a good place to mention a result of P.M. Neumann on epimorphisms. Namely, he proved that in any category of solvable groups which is closed under quotients, all epimorphisms are surjective. The result was later extended by S. McKay. See: Splitting groups and projectives, Peter M. Neumann, Quart. J. Math. Oxford (2), 18 (1967), 325-332; and Surjective epimorphisms in classes of groups, S. McKay, Quart. J. Math. Oxford (2) 20 (1969), 87-90.

   However, since not every amalgam of solvable groups is embeddable, and known conditions for embeddability are hard to work with, sundry categories of solvable groups seem like a worthwhile place to study dominions. In particular, we pay attention to the varieties of all solvable groups of length at most k, and to the category (pseudovariety) of all solvable groups.