derived series

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

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

in which G k=[G k1,G k1]G_k = [G_{k-1},G_{k-1}] is the commutator, that is the subgroup of G k1G_{k-1} generated by all elements of the form ghg 1h 1ghg^{-1}h^{-1} where g,hG k1g,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 LL, and for Ω\Omega-groups.

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