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An analogue of the Nielsen-Schreier formula for pro-\(p\)-groups. (English) Zbl 1119.20035

Let \(p\) be prime and \(G\) a finitely generated pro-\(p\) group. It is well known that \(G\) is a free pro-\(p\) group if and only if the Nielsen-Schreier formula \(d(U)-1=[G:U](d(G)-1)\) holds for all open subgroups \(U\) of \(G\), where \(d(U)\) denotes the minimal number of topological generators of \(U\). More in general one can consider the class \(\mathcal E_n\) of pro-\(p\)-groups \(G\) such that \(d(U)-n=[G:U](d(G)-n)\) for any open subgroup \(U\). The class \(\mathcal E_2\) contains all Demushkin groups and \(\mathbb{Z}_p^n\in\mathcal E_n\). The question is open whether there are other examples of groups in \(\mathcal E_n\).
In this paper, the author considers the inequality \(d(U)-n\leq [G:U](d(G)-n)\) instead, and studies \(i(G)\), the maximum of \(n\) such that the previous inequality holds for all open subgroups of \(G\). For example, he proves that, if \(|G|\geq 3\), then \(i(G)\leq\max\{d(G),r(G)+1\}\), \(r(G)\) being the minimal number of relations of \(G\) and gives information on the cases when either \(i(G)=d(G)\) or \(i(G)=r(G)+1\). Finally, he consider some questions about the Galois group of the maximal pro-\(p\)-extension of a number field with given ramification.

MSC:

20E18 Limits, profinite groups
20F05 Generators, relations, and presentations of groups
20E07 Subgroup theorems; subgroup growth
11R32 Galois theory
20J05 Homological methods in group theory
11R34 Galois cohomology
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