Nilpotent Group

A nilpotent group is a group $G$ such that its lower central series terminates at the trivial subgroup: there exists $c\ge 1$ with $\gamma_{c+1}(G)=\{e\}$, where $\gamma_1(G)=G$ and $\gamma_{k+1}(G) = [\gamma_k(G),G]$ (the subgroup generated by commutators $x^{-1}y^{-1}xy$ with $x\in\gamma_k(G)$ and $y\in G$).

Equivalently, $G$ is nilpotent iff its upper central series reaches $G$ in finitely many steps; in particular, nilpotent groups have large centers and are always solvable .

Examples:

• Every abelian group is nilpotent (of class $1$).
• Every finite p-group is nilpotent.
• The dihedral group of order $8$ and the quaternion group $Q_8$ are nilpotent.
• (Non-example) $S_3$ is not nilpotent.