Summary: The Prüfer rank \(\operatorname{rk}(G)\) of a profinite group \(G\) is the supremum, across all open subgroups \(H\) of \(G\), of the minimal number of generators \(\operatorname{d}(H)\). It is known that, for any given prime \(p\), a profinite group \(G\) admits the structure of a \(p\)-adic analytic group if and only if \(G\) is virtually a pro-\(p\) group of finite rank. The dimension \(\operatorname{dim}\, G\) of a \(p\)-adic analytic profinite group \(G\) is the analytic dimension of \(G\) as a \(p\)-adic manifold; it is known that \(\operatorname{dim}\, G\) coincides with the rank \(\operatorname{rk}(U)\) of any uniformly powerful open pro-\(p\) subgroup \(U\) of \(G\).
Let \(\pi\) be a finite set of primes, let \(r \in \mathbb{N}\) and let \(\mathbf{r} = (r_p)_{p \in \pi}\), \(\mathbf{d} = (d_p)_{p \in \pi}\) be tuples in \(\{0, 1, \ldots,r\} \). We show that there is a single sentence \(\sigma_{\pi,r,\mathbf{r},\mathbf{d}}\) in the first-order language of groups such that for every pro-\( \pi\) group \(G\) the following are equivalent: (i) \( \sigma_{\pi,r,\mathbf{r},\mathbf{d}}\) holds true in the group \(G\), that is, \(G \models \sigma_{\pi,r,\mathbf{r},\mathbf{d}} \); (ii) \(G\) has rank \(r\) and, for each \(p \in \pi \), the Sylow pro-\(p\) subgroups of \(G\) have rank \(r_p\) and dimension \(d_p\).
Loosely speaking, this shows that, for a pro-\( \pi\) group \(G\) of bounded rank, the precise rank of \(G\) as well as the ranks and dimensions of the Sylow subgroups of \(G\) can be recognized by a single sentence in the basic first-order language of groups.