upper central series

《Abstract Algebra》 · 第 190 页 · Section 6.1「p-Groups, Nilpotent Groups, and Solvable Groups」

For any (finite or infinite) group $G$ define the following subgroups inductively: $Z_0(G)=1$, and $Z_{i+1}(G)$ is the subgroup of $G$ containing $Z_i(G)$ such that $Z_{i+1}(G)/Z_i(G)=Z(G/Z_i(G))$. The chain of subgroups $1=Z_0(G)\le Z_1(G)\le Z_2(G)\le\cdots$ is called the upper central series of $G$.