On small profinite groups (Q2406667)

A profinite group is called small if it has only finitely many open subgroups of index \(n\) for each positive integer \(n\). A Frattini cover of a profinite group \(G\) is a continuous epimorphism \(\varphi: K \to G\) such that \(\ker(\varphi)\) is contained in the Frattini subgroup \(\Phi(K)\). The first result proved in the paper states that the Frattini cover of a small group is small. A profinite group is called strongly complete if every of its subgroup of finite index is open. The second result of the paper is the following {Theorem.} Suppose \(G\) and \(H\) are elementarily equivalent, i.e. satisfy the same sentences in the language of groups. If one of them is strongly complete, then \(G\) and \(H\) are isomorphic.