If $G/Z(G)$ is finite, then $|G'| < \infty$

Question

Let $G$ be an infinite group. Suppose that $G/Z(G)$ is a finite group, where $$G^{\prime}= \left\langle xyx^{-1}y^{-1}\ \middle\vert\ x,y \in G\right\rangle.$$ Prove that $|G'| < \infty$.

Answer 1

This is false. Let $H$ be a perfect group such that $H' = H$ and $Z(H) \neq 1$ (for example, $SL_n(F)$ for suitable $n$ and $F$) Then for $G = H \oplus H \oplus H \oplus \cdots$ (infinite direct sum), we have $G' = G$ but $Z(G)$ is infinite.