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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.

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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.

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