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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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    $\begingroup$ 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 Holt
    Apr 5, 2016 at 17:19
  • $\begingroup$ @DerekHolt Cool! $\endgroup$
    – Igor Rivin
    Apr 5, 2016 at 18:17
  • $\begingroup$ The second question is interesting in the case $k=1$. Does there exist a group $G$ with $G \cong [G,G] < G$? $\endgroup$
    – Derek Holt
    Apr 6, 2016 at 8:17
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    $\begingroup$ To Derek Holt: If again we allow infinitely generated groups, then the free group of countable rank would answer the question. $\endgroup$
    – Al Tal
    Apr 6, 2016 at 9:30
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    $\begingroup$ @AlTal Right, this is why the case of $k>1$ is more interesting... $\endgroup$
    – Igor Rivin
    Apr 6, 2016 at 11:27

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