Dp-minimal profinite groups and valuations on the integers

We study dp-minimal infinite profinite groups that are equipped with a uniformly definable fundamental system of open subgroups. We show that these groups have an open subgroup $A$ such that either $A$ is a direct product of countably many copies of $\mathbb{F}_p$ for some prime $p$, or $A$ is of the form $A \cong \prod_p \mathbb{Z}_p^{α_p} \times A_p$ where $α_p < ω$ and $A_p$ is a finite abelian $p$-group for each prime $p$. Moreover, we show that if $A$ is of this form, then there is a fundamental system of open subgroups such that the expansion of $A$ by this family of subgroups is dp-minimal. Our main ingredient is a quantifier elimination result for a class of valued abelian groups. We also apply it to $(\mathbb{Z},+)$ and we show that if we expand $(\mathbb{Z},+)$ by any chain of subgroups $(B_i)_{i<ω}$, we obtain a dp-minimal structure. This structure is distal if and only if the size of the quotients $B_i/B_{i+1}$ is bounded.

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