Nielsen-Schreier-type theorems for algebras of formal power series in noncommuting indeterminates
We say that the group G is a finite-dimensional \({\mathbb{Q}}\)-group if it admits a normal series of finite length, the factors of which are isomorphic to the additive group of \({\mathbb{Q}}\). Let \({\mathfrak N}\) and \(\overset \leftarrow {\mathfrak N}\) be the classes of finite-dimensional \({\mathbb{Q}}\)-groups and pro-\({\mathbb{Q}}\)-groups respectively. The author describes the free objects of finite and countable ranks in \(\overset \leftarrow {\mathfrak N}\) and proves that every closed subgroup of a free pro-\({\mathbb{Q}}\)-group is itself a free pro-\({\mathbb{Q}}\)-group. A similar theorem holds for Lie algebras. The method is based on the same idea as the proof of the theorem about subgroups of free pro-p-groups. The difficulties relate in the first place to the absence of compactness. Secondly, the cohomology remains continuous even for finite-dimensional \({\mathbb{Q}}\)-groups, which are discrete, and it is necessary to prove that it reduces in fact to discrete cohomology. Reviewer's remarks: 1. It seems doubtful that the proofs of analogous results for \({\mathbb{Z}}\)-groups ''are almost precise repetitions of the proofs for the case of \({\mathbb{Q}}\)-groups.'' 2. It seems that the propositions 5, 6 are wrong, but they are not used in the proof of the main theorem.
scientific article; zbMATH DE number 4002369