Let $G$ be a group. The upper central series for $G$ is an ascending sequence of subgroups of $G$ defined as follows: Let $Z_0(G) = {e}$ and let $Z_1(G)$ be the center $Z(G)$. Then for each $i\geq 1$, define $Z_{i+1}(G)$ to be a subgroup of $G$ containing $Z_i(G)$ such that the quotient by normal quotient $Z_{i+1}(G)/Z_i(G) = Z(G/Z_i(G))$. Note that because the center is a normal subgroup, these quotient by normal quotients are well-defined. Also note that this series may or may not reach $G$.