nLab
lower central series

Given a group GG, its lower central series is the inductively defined descending sequence

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

in which G k=[G,G k1]G_k = [G, G_{k-1}] is the subgroup generated by all commutators ghg 1h 1g h g^{-1}h^{-1} where gGg\in G and hG k1h\in G_{k-1}.

For a nilpotent group, this series terminates in finitely many steps at the trivial subgroup and is the same length as the upper central series. It is the fastest descending central series.

Similarly, given a Lie algebra LL, its lower central series is the inductively defined descending sequence of Lie subalgebras L=L 0L 1L 2L = L_0\supset L_1\supset L_2\supset\ldots in which L k=[L,L k1]L_k = [L, L_{k-1}] is the Lie subalgebra generated by all commutators [l,h][l,h] where lLl\in L and hL k1h\in L_{k-1}.

Revised on June 26, 2014 22:07:54 by David Corfield (31.185.244.23)