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Does every element of the derived series of a pro-p group is also a pro-p group?

The problem reduces to showing that every element of the derived series is a closed subgroup...But is it always true?

Hope you'll be able to help me

Thanks !

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1 Answer 1

up vote 7 down vote accepted

It seems this can't be expected. From Simons' thesis (p. xii):

The derived group of any finitely generated profinite group is closed, but Roman'kov [29] has provided an example of a finitely generated pro-p group in which the second derived group is not closed.

I haven't Roman'kov's paper at hand yet and don't know about details of his example. The reference is:

Roman'kov: The width of verbal subgroups of solvable groups. Algebra i Logika 21(1),60-72(1982).

In contrast, the groups in the lower central series of a finitely generated profinite group are always closed. This is proved by Nikolov-Segal in this paper (Theorem 1.4).

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Thanks ! !!!!!! –  jason mfash May 12 '12 at 9:55

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