---
tag: 0XW54DH28V
kind: concept
name: Upper central series
msc: [20F14]
sources:
  - url: https://en.wikipedia.org/wiki/Central_series
    locator: Upper central series
---

## Summary

The upper central series of a group $G$ is the sequence $\{e\} = Z_0 \subseteq Z_1 \subseteq Z_2 \subseteq \cdots$, where $Z_1$ is the centre and $Z_{i+1}/Z_i$ is the centre of $G/Z_i$.
The group is nilpotent if and only if this series reaches $G$.