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Pablo
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Action of a profinite group

Let $G$ be a finitely generated profinite group, $p$ a prime number. Put $$ V = \prod_{i \in I} \mathbb{Z}_p$$ a (profinite) group equipped with the product topology (for convenience, $I$ may be assumed to be countable). Suppose that $G$ acts by continuous automorphisms on $V$ (this means that $G$ acts continuously on $V$ respecting its group law, i.e $V$ is a profinte $G$-module. I am mainly interested in actions coming from extensions).

  1. Must $V$ contain a closed, nontrivial, topologically finitely generated subgroup invariant under the action of $G$?

  2. Must $V$ contain a nontrivial pair of trivially intersecting $G$-submodules?

  3. Is there a way to decompose $V$ into a direct product of $G$-submodules?

I am equally interested in the case of $\mathbb{F}_p$ in place of $\mathbb{Z}_p$.

Pablo
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