Upper central series of a group (reviewed)

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 $i$.