Let $G$ be a finitely generated amenable group and $G^{(i)}$ be a derived subgroup of $G$.
What can we say about the center of $G^{(i)}$? Can we say the center of $G^{(i)}$ is non-trivial for $i$ large enough?
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6$\begingroup$ Juschenko-Monod recently constructed a f.g. simple infinite amenable group $G$. Clearly it has trivial center and its derived subgroups all equal $G$. $\endgroup$– Igor BelegradekCommented May 11, 2013 at 21:17
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1$\begingroup$ Grigorchuk group is amenable. One can show (by virtue of the branch property) that the center of every derived subgroup is trivial. $\endgroup$– Mustafa Gokhan BenliCommented May 11, 2013 at 21:59
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2$\begingroup$ Consider $S_5$. $\endgroup$– user6976Commented May 12, 2013 at 0:09
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2$\begingroup$ @Mahdi: You have not mentioned in your question that the requested group must be infinite, so Mark's answer is Ok. $\endgroup$– Alireza AbdollahiCommented May 12, 2013 at 1:22
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5$\begingroup$ There is no word "infinite" in the question. If you really want infinite, take the wreath product $S_5\wr {\mathbb Z}$. $\endgroup$– user6976Commented May 12, 2013 at 2:14
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