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

Abstract

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 prop 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 prop subgroup U of G.Let π be a finite set of primes, let r ∈ ‫ގ‬ and let r = (r p ) p∈π , d = (d p ) 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 prop subgroups of G have rank r p and dimension d p .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.