derived series

Given a group $G$ its derived series is the decreasing (under inclusion order), inductively defined sequence of its subgroups

$$G={G}_{0}\supset {G}_{1}\supset {G}_{2}\supset {G}_{3}\supset \dots $$

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 ${\mathrm{ghg}}^{-1}{h}^{-1}$ 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.

Created on June 16, 2011 18:17:20
by Zoran Škoda
(161.53.130.104)