Specifically, let be a group. The derived series is a sequence defined recursively as , , where is the derived group (i.e., the commutator subgroup) of a group .
A group for which is trivial for sufficiently large is called solvable. The least such that is called the solvability class of . By transfinite recursion, this notion can be extended to infinite ordinals, as well.
Let be the th term of the lower central series of . Then from the relation and induction, we have In particular, if is nilpotent of class at most , then it is solvable of class at most . Thus if is nilpotent, then it is solvable; however, the converse is not generally true.
If is a decreasing sequence of subgroups such that is a normal subgroup of and is abelian for all integers , then , by induction on .