A group is nilpotent exactly when its upper central series reaches the whole group.Suggest a slogan
Let \(G\) be a group with upper central series \(Z_0\leq Z_1\leq\cdots\). Then \(G\) is nilpotent if and only if \(Z_n=G\) for some \(n\geq 0\).
Nilpotent groups via the upper central series
T-7PU33