Show that every abelian group is nilpotent.
Solution
Let \(G\) be abelian. Then \[ [G,G]=1. \] The lower central series is \[ \gamma_1(G)=G, \qquad \gamma_{i+1}(G)=[\gamma_i(G),G]. \] Therefore \[ \gamma_2(G)=[G,G]=1. \] So the lower central series reaches the identity after one nontrivial step. Hence \(G\) is nilpotent of class at most \(1\).
Equivalently, since \(G\) is abelian, \[ Z(G)=G, \] so the upper central series reaches all of \(G\) immediately.