Abstract
We intend to generalize a crucial lemma of [4] to prove a somewhat surprising arithmetic property of profinite groups; namely, that a profinite group G has nontrivial p-Sylow-subgroups for only a finite number of primes if and only if this is true for its procyclic subgroups. This will yield as a corollary that every profinite torsion group has finite exponent if and only if this is true for its Sylow-sub-groups, a result also contained in [4].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
BERGER T.R.: Automorphisms of Solvable Groups. Journal of Algebra27, 311 (1973)
GROSS F.: Solvable groups admitting a fixed-point-free automorphism of prime power order. Proc.Amer.Math.Soc.17, 1440–1446 (1966)
GILDENHUYS D., HVERFORT W., RIBES L.: Profinite Frobenius-groups. Arch.d.Math.33, 518–528 (1979)
HVERFORT W.: Compact Torsion groups and finite exponent. Arch.d.Math.33, 404–410 (1979)
HUPPERT B.: Endliche Gruppen I. Berlin Heidelberg New York 1967
RIBES L.: Introduction to profinite groups and Galois cohomology. Queen's Papers in Pure and Applied Mathematics 24, Queen's University, Kingston, 1970
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Herfort, W. An arithmetic property of profinite groups. Manuscripta Math 37, 11–17 (1982). https://doi.org/10.1007/BF01239941
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01239941