---
tag: 0DJ9Y9V610
kind: definition
name: Upper central series
aliases: [ascending-central-series]
msc: [20F14]
defines: 0XW54DH28V
uses: [0S7Y9Q990Q, 0YYWCNTPWR, 0S9ZVZ34MD]
sources:
  - url: https://en.wikipedia.org/wiki/Central_series
    locator: Upper central series
---

## Definition

Let $G$ be a [[group|G5S1M9]].
The upper central series of $G$ is the sequence of normal subgroups $Z_0 = \{e\}$ and $Z_{i+1} = \{x \in G : [x, y] \in Z_i \text{ for all } y \in G\}$ for $i \geq 0$.

## Remarks

The first term $Z_1$ is the [[centre|0DZ93493VN]] $Z(G)$, and $Z_2$ is called the second centre.
For each $i$, $Z_{i+1}/Z_i$ is the centre of the [[quotient group|0KYX2N69J0]] $G/Z_i$.
For the quaternion group $Q_8$, the series is $\{1\} \subseteq \{1, -1\} \subseteq Q_8$.
A group with trivial centre, such as $S_3$, has $Z_i = \{e\}$ for each $i$.

## Explanations

### L1 human

The upper central series starts with the identity and first adds the centre.
Then it adds the elements that are central modulo the centre, and so on.
If this process reaches the whole group, the group is nilpotent.