  A profinite group is called small if it has only finitely many open subgroups
of index n for each positive integer n. We show that every Frattini cover of a
small profinite group is small. A profinite group is called strongly complete
if every subgroup of finite index is open. We show that two profinite groups
that are elementarily equivalent, in the first-order language of groups, are
isomorphic if one of them is strongly complete, extending a result of Moshe
Jarden and Alexander Lubotzky which treats the case of finitely generated
profinite groups.
