Show that every abelian group is nilpotent.
Solution
If \(G\) is abelian, then every commutator is trivial: \[ [x,y]=xyx^{-1}y^{-1}=1. \] Hence \[ [G,G]=1. \] Therefore the lower central series \[ G\supseteq [G,G]\supseteq\cdots \] reaches the trivial subgroup after one step. Thus every abelian group is nilpotent, of nilpotency class at most \(1\).