# nLab lower central series

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

$G={G}_{0}\supset {G}_{1}\supset {G}_{2}\supset \dots$G = G_0 \supset G_1\supset G_2\supset \ldots

in which ${G}_{k}=\left[G,{G}_{k-1}\right]$ is the subgroup generated by all commutators ${\mathrm{ghg}}^{-1}{h}^{-1}$ where $g\in G$ and $h\in {G}_{k-1}$.

Similarly, given a Lie algebra $L$, its lower central series is the inductively defined descending sequence of Lie subalgebras $L={L}_{0}\supset {L}_{1}\supset {L}_{2}\supset \dots$ in which ${L}_{k}=\left[L,{L}_{k-1}\right]$ is the Lie subalgebra generated bt all commutators $\left[l,h\right]$ where $l\in L$ and $h\in {L}_{k-1}$.