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).

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

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

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

I am equally interested in the case of $\mathbb{F}_p$ (the field of cardinality $p$) in place of $\mathbb{Z}_p$ (the $p$-adic integers).

2more comments