The Prüfer rank
$\mathrm{rk}(G)$ 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
$\mathrm{dim}G$
of a
$p$adic analytic
profinite group
$G$ is the
analytic dimension of
$G$ as a
$p$adic manifold; it is known
that
$\mathrm{dim}G$ coincides with the
rank
$\mathrm{rk}(U)$ 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
$r={({r}_{p})}_{p\in \pi},d={({d}_{p})}_{p\in \pi}$ be tuples in
$\{0,1,\dots ,r\}$. We show that there
is a single sentence
${\sigma}_{\pi ,r,r,d}$
in the firstorder language of groups such that for every
pro$\pi $ group
$G$ the following are
equivalent: (i)
${\sigma}_{\pi ,r,r,d}$ holds
true in the group
$G$,
that is,
$G\vDash {\sigma}_{\pi ,r,r,d}$; (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 firstorder language of
groups.
