Say a topological group $G$ is topologically characteristically simple if there does not exist a closed subgroup $1 < K < G$ such that $K$ is invariant under all automorphisms of $G$ (here `automorphisms' are required to preserve the topology as well as the group structure).

Q1: Has someone written down a classification of locally compact second-countable abelian groups that are topologically characteristically simple?

I tried to derive a classification myself; the case I am having difficulty with is when $G$ is torsion-free and the group $pG$ of $p$-th powers in $G$ is a proper dense subgroup of $G$. (Another possible division of this case is between those $G$ with a dense divisible subgroup, and those with no divisible subgroup, but I don't know where to go from there.)

I know one such group for each $p$: let $G$ be the group of all functions from $\mathbb{N}$ to $\mathbb{Q}_p$ under pointwise addition such that all but finitely many values are in $\mathbb{Z}_p$, and topologise it so that the group of functions from $\mathbb{N}$ to $\mathbb{Z}_p$ is an open subgroup with the compact-open topology. This example is topologically characteristically simple with a dense divisible subgroup (the finitely supported functions).

Q2: Are there any other locally compact second-countable abelian groups $G$ such that $G$ is torsion-free, $pG$ is a proper dense subgroup (edit: and $x^{p^n} \rightarrow 1$ for all $x \in G$)? If so, are any of them topologically characteristically simple?

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    $\begingroup$ In the first part of Q2 there's a trivial answer, e.g. the direct product of your $G$ (I met this group a couple of times, I don't remember where of if I used it anywhere) and the group of rationals, so probably only the TCS assumption makes the question nontrivial. An equivalent variant is to ask about classifying TCS locally compact groups with an open subgroup isomorphic to $\mathbf{Z}_p^\mathbf{N}$. $\endgroup$ – YCor Apr 24 '16 at 22:37
  • $\begingroup$ Ah yes, I meant to exclude trivial examples like that (e.g. by also requiring $G$ to be locally elliptic). Indeed it is the TCS condition that gives some hope of a classification. It looks like all examples will have a compact open subgroup $U$ isomorphic to $\mathbb{Z}^{\mathbb{N}}_p$ with $G/U$ isomorphic to a direct sum of Prüfer $p$-groups, but there could be many ways of putting $U$ and $G/U$ together. $\endgroup$ – Colin Reid Apr 25 '16 at 6:11
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    $\begingroup$ Or at least assume that $G$ is $p$-elliptic (i.e., is a $\mathbf{Z}_p$-module, or, equivalently but apparently more intrinsically, that $p^nx\to 0$ when $n\to\infty$ for all $x$). $\endgroup$ – YCor Apr 25 '16 at 6:35

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