On subgroups of finite index in compact Hausdorff groups.
Let \(G\) be a compact Hausdorff group. The authors prove that all abstract subgroups of finite index in \(G\) are open if and only if \(G\) has only finitely many abstract subgroups of index \(n\) for each integer \(n\) if and only if \(G\) has only countably many abstract subgroups of finite index. The proof uses results concerning varieties generated by finite groups and the authors also prove that the statements above are equivalent to the following. Suppose \(V\) is a verbal subgroup of \(G\) corresponding to the variety generated by a finite group then \(V\) is open. The authors remark that some of the interest in results such as these has been generated by J.-P. Serre's well-known conjecture, that in a (topologically) finitely-generated profinite group every subgroup of finite index is open. We note that \textit{N. Nikolav} and \textit{D. Segal} have recently announced a proof of Serre's conjecture [C. R., Math., Acad. Sci. Paris 337, No. 5, 303--308 (2003; Zbl 1033.20029)].
scientific article; zbMATH DE number 1966434