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The structure of some virtually
free pro-$p$ groups


Author: Claus Scheiderer
Journal: Proc. Amer. Math. Soc. 127 (1999), 695-700
MSC (1991): Primary 20E18; Secondary 20E34, 20E36, 20E06
DOI: https://doi.org/10.1090/S0002-9939-99-04765-6
MathSciNet review: 1487337
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Abstract: We prove two conjectures on pro-$p$ groups made by Herfort, Ribes and Zalesskii. The first says that a finitely generated pro-$p$ group which has an open free pro-$p$ subgroup of index $p$ is a free pro-$p$ product $H_0*(S_1\times H_1)*\cdots*(S_m\times H_m)$, where the $H_i$ are free pro-$p$ of finite rank and the $S_i$ are cyclic of order $p$. The second says that if $F$ is a free pro-$p$ group of finite rank and $S$ is a finite $p$-group of automorphisms of $F$, then $\operatorname{Fix}(S)$ is a free factor of $F$. The proofs use cohomology, and in particular a ``Brown theorem'' for profinite groups.


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Additional Information

Claus Scheiderer
Affiliation: Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
Email: claus.scheiderer@mathematik.uni-regensburg.de

DOI: https://doi.org/10.1090/S0002-9939-99-04765-6
Keywords: Pro-$p$ groups, virtually free groups, group cohomology, Brown theorem
Received by editor(s): July 1, 1997
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 1999 American Mathematical Society

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