derived series

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

G=G 0G 1G 2G 3G = G_0 \supset G_1 \supset G_2\supset G_3\supset \ldots

in which G k=[G k1,G k1] is the commutator, that is the subgroup of G k1 generated by all elements of the form ghg 1h 1 where g,hG k1. 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 Ω-groups.

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