Given a group its derived series is the decreasing (under inclusion order), inductively defined sequence of its subgroups
G = G_0 \supset G_1 \supset G_2\supset G_3\supset \ldots
in which is the commutator, that is the subgroup of generated by all elements of the form where . A group is solvable iff its derived series terminates with the trivial subgroup after finitely many terms.
Similarly, one defines a derived series for a Lie algebra , and for -groups.