Nilpotent group

Definition

A group $G$ is nilpotent when it has a central series. This is a finite sequence $\{e\} = N_0 \subseteq N_1 \subseteq \cdots \subseteq N_n = G$ of normal subgroups of $G$ with $N_{i+1}/N_i \subseteq Z(G/N_i)$ for each $i < n$. Here $Z(G/N_i)$ is the centre of the quotient group $G/N_i$.

Remarks

The condition $N_{i+1}/N_i \subseteq Z(G/N_i)$ means $[N_{i+1}, G] \subseteq N_i$. Equivalently, the lower central series reaches $\{e\}$, or the upper central series reaches $G$; the least such length is the nilpotency class. The nilpotent groups of class at most $1$ are exactly the abelian groups. The quaternion group $Q_8$ is nilpotent of class $2$, but $S_3$ is not nilpotent.

Explanations

L1 human

A nilpotent group is close to abelian in a strong sense. Each layer of a central series commutes with the whole group, up to the lower layers. Each nilpotent group is solvable, but the converse is false, for example for $S_3$.