Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 7 / 1 / i / Solution

A central series is a finite chain of normal subgroupswith $G_i/G_{i-1}\le Z(G/G_{i-1})$ for every $i\ge 1$. Equivalently, writing $[x,y]=x^{-1}{y}^{-1}xy$ and $[A,B]$ for the subgroup generated by these group commutators, the condition is $[G_i,G]\le G_{i-1}$. Centrality is relative to the whole quotient group, not just to the next subgroup in the chain.

A group is nilpotent if it admits a finite central series. The trivial group admits a series of length zero. For a nontrivial nilpotent group, the minimum possible length is its nilpotency class.