Derived Series of Pro-p groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T13:23:38Z http://mathoverflow.net/feeds/question/96750 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96750/derived-series-of-pro-p-groups Derived Series of Pro-p groups jason mfash 2012-05-12T07:27:44Z 2012-05-12T09:56:10Z <p>Does every element of the derived series of a pro-p group is also a pro-p group?</p> <p>The problem reduces to showing that every element of the derived series is a closed subgroup...But is it always true?</p> <p>Hope you'll be able to help me</p> <p>Thanks ! </p> http://mathoverflow.net/questions/96750/derived-series-of-pro-p-groups/96756#96756 Answer by Ralph for Derived Series of Pro-p groups Ralph 2012-05-12T09:50:44Z 2012-05-12T09:50:44Z <p>It seems this can't be expected. From <a href="http://ora.ouls.ox.ac.uk/objects/uuid%253A01075c36-c7e6-4def-9647-86b4346e4726/datastreams/THESIS01" rel="nofollow">Simons' thesis</a> (p. xii): </p> <blockquote> <p>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. </p> </blockquote> <p>I haven't Roman'kov's paper at hand yet and don't know about details of his example. The reference is: </p> <p>Roman'kov: The width of verbal subgroups of solvable groups. Algebra i Logika 21(1),60-72(1982). </p> <p>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 <a href="http://annals.math.princeton.edu/wp-content/uploads/annals-v165-n1-p05.pdf" rel="nofollow">this paper </a> (Theorem 1.4). </p>