Elementary abelian normal subgroups of a pro-p group - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T06:39:07Z http://mathoverflow.net/feeds/question/30493 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/30493/elementary-abelian-normal-subgroups-of-a-pro-p-group Elementary abelian normal subgroups of a pro-p group Colin Reid 2010-07-04T09:07:03Z 2010-07-27T15:53:17Z <p>Let \$P\$ be a (finitely generated) pro-\$p\$ group, and let \$E\$ be an infinite elementary abelian normal subgroup. Does \$E\$ necessarily contain a non-trivial finite normal subgroup of \$P\$? We can think of \$E\$ as consisting of sequences of elements of \$C_p\$, with open subgroups \$O_X\$, where \$X\$ is a finite subset of the indexing set and \$O_X\$ consists of the sequences that are zero on \$X\$. However, I can't think of a way of making \$P\$ act on these sequences that doesn't leave some finite subgroup invariant. Acting on the indexing set is no good because the orbits would have to be finite, and you'd have a finite normal subgroup consisting of sequences that are zero outside some given orbit.</p> http://mathoverflow.net/questions/30493/elementary-abelian-normal-subgroups-of-a-pro-p-group/33536#33536 Answer by Colin Reid for Elementary abelian normal subgroups of a pro-p group Colin Reid 2010-07-27T15:53:17Z 2010-07-27T15:53:17Z <p>After talking to Charles Leedham-Green, I now have an example that answers the question (I think). See <a href="http://mathoverflow.net:80/questions/33533/name-this-pro-p-group" rel="nofollow">http://mathoverflow.net:80/questions/33533/name-this-pro-p-group</a>. More interesting examples would still be nice though, particularly if they do not have \$C_p \wr C_p\$ as an image.</p>