Let \(G\) be a group. Define normal subgroups \(Z_i\normal G\) inductively by \(Z_0\coloneqq 1\) and, for \(i\geq 0\), \[ Z_{i+1}\coloneqq \pi_i^{-1}\bigl(Z(G/Z_i)\bigr), \] where \(\pi_i\colon G\to G/Z_i\) is the quotient map and \(Z(G/Z_i)\) is the center of \(G/Z_i\). The upper central series of \(G\) is the chain \(Z_0\leq Z_1\leq Z_2\leq\cdots\).
The induction is well defined: if \(Z_i\normal G\), then \(Z(G/Z_i)\) is normal in \(G/Z_i\), so its preimage \(Z_{i+1}\) is normal in \(G\). \(Z_{i+1}\) is the unique subgroup of \(G\) containing \(Z_i\) with \(Z_{i+1}/Z_i=Z(G/Z_i)\); in particular \(Z_1=Z(G)\).