nLab lower central series

Idea

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

$$G \,=\, G_0 \supset G_1\supset G_2\supset \ldots$$

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

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 $L$, its lower central series is the inductively defined descending sequence of Lie subalgebras $L = L_0\supset L_1\supset L_2\supset\ldots$ in which $L_k = [L, L_{k-1}]$ is the Lie subalgebra generated by all commutators $[l,h]$ where $l\in L$ and $h\in L_{k-1}$.

Related concepts

• upper central series

• completion of a group

References

See also

• Wikipedia, Central series