Small profinite groups (Q2747725)

A profinite structure is a profinite topological space (the inverse limit of a system of finite discrete spaces) together with a suitably defined group of automorphisms acting on it. A model-theoretic approach to profinite structures (including appropriate notions of definability, interpretability, smallness, stability -- the so called \(m\)-stability -- and a related notion of rank -- the \({\mathcal M}\)-rank) were developed by the author (and others) in some previous papers. Here this program is pursued to study small profinite groups. These can be introduced as profinite groups \(G\), hence inverse limits of some inverse system \(\overline G = (G_i, f_{i,j})_{i,j \in I}\) (with \(I\) countable), such that the (also profinite) group of the automorphisms of \(G\) induced by automorphisms of \(\overline G\) has at most countably many orbits on \(G^n\) for every positive integer \(n\). But they can also be viewed as small profinite structures with a definable group law. Any profinite group interpetable in a small theory \(T\) is small. NEWLINENEWLINENEWLINEIn a previous paper the author singled out some model-theoretic assumptions on a theory \(T\) sufficient to ensure that any small profinite group interpretable in \(T\) contains an open abelian subgroup. So a natural question arises: Does every small profinite group contain an open abelian subgroup? The author investigates this problem within the model-theoretic framework sketched before and, although he conjectures a negative general solution, he answers positively some partial settings, in particular the easiest case when the profinite group is just a product of finite groups. The related question whether every profinite group interpretable in a small \(T\) contains an open abelian subgroup is also discussed. Indeed, it is observed that the latter question is equivalent to the former one.