Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question

1 Answer 1

up vote 1 down vote accepted

After talking to Charles Leedham-Green, I now have an example that answers the question (I think). See http://mathoverflow.net:80/questions/33533/name-this-pro-p-group. More interesting examples would still be nice though, particularly if they do not have $C_p \wr C_p$ as an image.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.