Finite axiomatizability of the rank and the dimension of a pro-π group

The Prüfer rank rk (G) of a profinite group G is the supremum, across all open subgroups H of G, of the minimal number of generators 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 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 dim G coincides with the rank rk (U) of any uniformly powerful open pro-p subgroup U of G. Let π be a finite set of primes, let r ∈ N and let r = (rp) p∈π, d = (dp) p∈π be tuples in 0, 1,. . . , r. We show that there is a single sentence σ π, r, r, d in the first-order language of groups such that for every pro-π group G the following are equivalent: (i) σ π, r, r, d holds true in the group G, that is, G |= σ π, r, r, d ; (ii) G has rank r and, for each p ∈ π, the Sylow pro-p subgroups of G have rank rp and dimension dp. Loosely speaking, this shows that, for a pro-π 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.