---
tag: 08PA437800
kind: definition
name: Lower central series
aliases: [descending-central-series]
msc: [20F14]
defines: 0K98R7A4VC
uses: [06RGT4V6Y6, 0RG9SPQSWM, 08SJ5K2CBG]
sources:
  - url: https://en.wikipedia.org/wiki/Central_series
    locator: Lower central series
---

## Definition

Let $G$ be a [[group|G5S1M9]].
For subgroups $H$ and $K$ of $G$, let $[H, K]$ be the subgroup [[generated by|0RP7VWQ0YN]] the [[commutators|0PXW0BN7DW]] $[h, k]$ with $h \in H$ and $k \in K$.
The lower central series of $G$ is the sequence $\gamma_1(G) = G$ and $\gamma_{i+1}(G) = [\gamma_i(G), G]$ for $i \geq 1$.

## Remarks

The second term $\gamma_2(G) = [G, G]$ is the [[commutator subgroup|0ASGSTXSBW]].
Each $\gamma_i(G)$ is normal in $G$, and $\gamma_i(G)/\gamma_{i+1}(G)$ lies in the centre of $G/\gamma_{i+1}(G)$.
The terms of the [[derived series|0Q5G5DJ4P2]] satisfy $G^{(i)} \subseteq \gamma_{i+1}(G)$.
For $S_3$, the series is $S_3 \supseteq A_3 \supseteq A_3 \supseteq \cdots$, so it never reaches $\{e\}$.

## Explanations

### L1 human

The lower central series starts with the whole group.
At each step, it takes the commutators of the last term with all elements of the group.
If this process reaches the identity, the group is nilpotent.