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.

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