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Cohomology and finite subgroups of profinite groups

Authors: Pham Anh Minh and Peter Symonds
Journal: Proc. Amer. Math. Soc. 132 (2004), 1581-1588
MSC (2000): Primary 20J06, 17B50
Published electronically: November 4, 2003
MathSciNet review: 2051117
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove two theorems linking the cohomology of a pro-$p$ group $G$ with the conjugacy classes of its finite subgroups.

The number of conjugacy classes of elementary abelian $p$-subgroups of $G$ is finite if and only if the ring $H^{*}(G,\mathbb{Z} /p)$ is finitely generated modulo nilpotent elements.

If the ring $H^{*}(G,\mathbb{Z} /p)$ is finitely generated, then the number of conjugacy classes of finite subgroups of $G$ is finite.

References [Enhancements On Off] (What's this?)

  • 1. K. S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, Berlin, 1982. MR 83k:20002
  • 2. J. D. Dixon, M.P.F. du Sautoy, A. Mann, and D. Segal, Analytic pro-$p$ groups (2nd edition), London Mathematical Society Lecture Note Series, vol. 157, Cambridge University Press, Cambridge, 1999. MR 94e:20037
  • 3. W. G. Dwyer and C. Wilkerson, Smith Theory and the functor $T$, Comment. Math. Helvetici 66 (1991), 1-17. MR 92i:55006
  • 4. L. Evens, The cohomology of groups, Oxford Mathematical Monographs, Clarendon Press, 1991. MR 93i:20059
  • 5. H.-W. Henn, Centralizers of elementary abelian $p$-subgroups and mod-$p$ cohomology of profinite groups, Duke Math. J. 91 (1998), 561-585. MR 99b:20083
  • 6. H.-W. Henn, J. Lannes, and L. Schwarz, The categories of unstable modules and unstable algebras over the Steenrod algebra modulo nilpotent objects, Amer. Jour. Math. 115 (1993), 1053-1106. MR 94i:55024
  • 7. D. Quillen, A cohomological criterion for $p$-nilpotence, J. Pure Appl. Algebra 1 (1971), 361-372. MR 47:6886
  • 8. D. Quillen, The spectrum of an equivariant cohomology ring II, Ann. Math. 94 (1971), 573-602. MR 45:7743
  • 9. C. Scheiderer, Farrell cohomology and Brown theorems for profinite groups, Manuscripta Math. 91 (1996), 247-281. MR 97j:20050

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Additional Information

Pham Anh Minh
Affiliation: Department of Mathematics, College of Science, University of Hue, Dai hoc Khoa hoc, Hue, Vietnam
Address at time of publication: Inst. Hautes Études Sci., Le Bois-Marie, 35 Route de Chartres, F-91440 Bures-sur-Yvette, France

Peter Symonds
Affiliation: Department of Mathematics, U.M.I.S.T., P.O. Box 88, Manchester M60 1QD, England

Received by editor(s): November 1, 2002
Received by editor(s) in revised form: February 9, 2003
Published electronically: November 4, 2003
Communicated by: Stephen D. Smith
Article copyright: © Copyright 2003 American Mathematical Society

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