Welcome to MathOverflow

Question

3

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 !

edited May 12 2012 at 7:36

asked May 12 2012 at 7:27

jason mfash
215●1●4

Ralph · Accepted Answer · 2012-05-12 09:50:44Z UTC

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).

Derived Series of Pro-p groups

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.