If $G$ is a group, then its **upper central series** is a normal series of subgroups
$$ \langle e\rangle = Z_0(G) \unlhd Z_1(G) \unlhd Z_2(G) \unlhd \cdots $$
where $Z_0(G)=\langle e\rangle$ and $Z_i(G)/Z_{i-1}(G)$ is the center of $G/Z_{i-1}(G)$, i.e., $Z_i(G)$ is the inverse image of the center of $G/Z_{i-1}(G)$.

Each of the subgroups $Z_i(G)$ is characteristic. The group is nilpotent if and only if $Z_i(G)=G$ for some $i$.

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**Knowl status:**

- Review status: beta
- Last edited by John Jones on 2019-06-12 14:33:25

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