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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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