## If a group mod its center is nilpotent, then the group is nilpotent

Prove that if $G/Z(G)$ is nilpotent, then $G$ is nilpotent.

We proved, as a lemma to the previous exercise, that $Z_t(G/Z(G)) = Z_{t+1}(G)/Z(G)$ for all $t$. Now suppose $G/Z(G)$ is nilpotent of nilpotence class $k$. Then $G/Z(G) = Z_k(G/Z(G)) = Z_{k+1}(G)/Z(G)$, and we have $G = Z_{k+1}(G)$. Thus $G$ is nilpotent of nilpotence class at most $k+1$.