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On a new characterization of Demuskin groups. (English) Zbl 0546.20021
The key result is Theorem 2. Let \(G\) be a finitely generated one-relator pro-\(p\)-group of rank \(n\neq 1\). If \(G\) is not a Demuskin group, then there exists an open subgroup \(U\) of \(G\) of index \(p\) with \(\dim H^ 2(U)>1\) and \(n(U)-2>p(n-2)\). The proof is based on a normal form of the relator and uses Lazard’s technique of \((x,\tau,p)\)-filtrations of a free pro-\(p\)-group and the five-term cohomology exact sequence of a group extension [cf. the second author, Can. J. Math. 19, 106-132 (1967; Zbl 0153.04202)]. Under the same hypotheses as Theorem 2, the following are equivalent: (a) The group \(G\) is a Demuskin group. (b) Every open subgroup of index \(p\) in \(G\) is a one-relator group. (c) If \(U\) is an open subgroup (of index \(p\)) in \(G\), then \(n(U)-2=[G:U](n(G)-2)\). Statement (c) provides a partial answer to a question asked by Iwasawa where the above result deals with the case \(c=2\).
Reviewer: H.R.Schneebeli

MSC:
20E18 Limits, profinite groups
20J05 Homological methods in group theory
20F05 Generators, relations, and presentations of groups
20F40 Associated Lie structures for groups
20E07 Subgroup theorems; subgroup growth
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References:
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