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 $G$.