Consider a group $G.$ Is it possible for $G$ to have a subnormal series $G \triangleright G_1 \triangleright \dotsc \triangleright G_n \triangleright \dotsc$ which cycles - that is, with $G_{i+k} \simeq G_i,$ for some $k>1?$ To make the problem more rigid, same question, but with the derived series instead of a subnormal series.
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5$\begingroup$ For the first question, start with a direct product of countably infinitely many copies of a cyclic group of order $2^k$ and then descend in steps of index $2$ going down through each direct factor in turn. It's even a normal series! $\endgroup$– Derek HoltApr 5, 2016 at 17:19
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$\begingroup$ @DerekHolt Cool! $\endgroup$– Igor RivinApr 5, 2016 at 18:17
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$\begingroup$ The second question is interesting in the case $k=1$. Does there exist a group $G$ with $G \cong [G,G] < G$? $\endgroup$– Derek HoltApr 6, 2016 at 8:17
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1$\begingroup$ To Derek Holt: If again we allow infinitely generated groups, then the free group of countable rank would answer the question. $\endgroup$– Al TalApr 6, 2016 at 9:30
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1$\begingroup$ @AlTal Right, this is why the case of $k>1$ is more interesting... $\endgroup$– Igor RivinApr 6, 2016 at 11:27
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