derived series

Given a group $G$ 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 $G_k = [G_{k-1},G_{k-1}]$ is the commutator, that is the subgroup of $G_{k-1}$ generated by all elements of the form $ghg^{-1}h^{-1}$ where $g,h\in G_{k-1}$. 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 $L$, and for $\Omega$-groups.