- group, ∞-group
- group object, group object in an (∞,1)-category
- abelian group, spectrum
- super abelian group
- group action, ∞-action
- representation, ∞-representation
- progroup
- homogeneous space

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 $h\in L_{k-1}$.

See also

- Wikipedia,
*Central series*

Last revised on July 16, 2022 at 17:52:25. See the history of this page for a list of all contributions to it.