Lower central series

Let $G$ be a group . The lower central series of $G$ is the sequence of subgroups $(\gamma_n(G))_{n\ge 1}$ defined by $\gamma_1(G)=G,\qquad \gamma_{n+1}(G)=[\gamma_n(G),G]\ \ (n\ge 1),$ where for subgroups $A,B\le G$ one writes $[A,B]=\langle [a,b] : a\in A,\ b\in B\rangle$ for the subgroup generated by all commutators $[a,b]=a^{-1}b^{-1}ab$ with $a\in A$ and $b\in B$.

The series is descending: $\gamma_{n+1}(G)\le \gamma_n(G)$, and $\gamma_n(G)\lhd G$ for all $n$. Note that $\gamma_2(G)=[G,G]$ is the commutator subgroup . A group is nilpotent if and only if $\gamma_{c+1}(G)$ is the trivial subgroup for some $c\ge 1$; the least such $c$ is the nilpotency class.

Examples:

• If $G$ is abelian, then $\gamma_2(G)=\{e\}$, so the lower central series terminates immediately.
• For $S_3$, one has $\gamma_2(S_3)=A_3$, and in fact $\gamma_n(S_3)=A_3$ for all $n\ge 2$, so the series does not reach $\{e\}$; hence $S_3$ is not nilpotent.
• For the dihedral group $D_8$ of order $8$, the lower central series terminates after finitely many steps (indeed $D_8$ is nilpotent).