Upper central series

Let $G$ be a group . The upper central series of $G$ is the sequence of subgroups $(Z_n(G))_{n\ge 0}$ defined by $Z_0(G)=\{e\},\qquad Z_1(G)=Z(G),$ where $Z(G)$ is the center of $G$, and for $n\ge 0$ one defines $Z_{n+1}(G)$ to be the preimage of the center of the quotient group $G/Z_n(G)$ under the natural projection $G\to G/Z_n(G)$. Equivalently, $Z_{n+1}(G)/Z_n(G)=Z\bigl(G/Z_n(G)\bigr).$

Each $Z_n(G)$ is a characteristic subgroup of $G$ (hence normal). A group is nilpotent if and only if $Z_c(G)=G$ for some $c\ge 0$; the least such $c$ is the nilpotency class.

Examples:

• If $G$ is abelian, then $Z_1(G)=G$, so the upper central series reaches $G$ immediately.
• For $S_3$, the center is trivial, so $Z_n(S_3)=\{e\}$ for all $n$; hence $S_3$ is not nilpotent.
• For $D_8=\langle r,s\mid r^4=s^2=e,\ srs=r^{-1}\rangle$, one has $Z_1(D_8)=\{e,r^2\}$ and $Z_2(D_8)=D_8$, so $D_8$ is nilpotent of class $2$.