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Abelian subgroups of topological groups


Authors: Siegfried K. Grosser and Wolfgang N. Herfort
Journal: Trans. Amer. Math. Soc. 283 (1984), 211-223
MSC: Primary 22A05; Secondary 22D05
DOI: https://doi.org/10.1090/S0002-9947-1984-0735417-4
MathSciNet review: 735417
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Abstract: In [1] Šmidt's conjecture on the existence of an infinite abelian subgroup in any infinite group is settled by counterexample. The well-known Hall-Kulatilaka Theorem asserts the existence of an infinite abelian subgroup in any infinite locally finite group. This paper discusses a topological analogue of the problem. The simultaneous consideration of a stronger condition--that centralizers of nontrivial elements be compact--turns out to be useful and, in essence, inevitable. Thus two compactness conditions that give rise to a profinite arithmetization of topological groups are added to the classical list (see, e.g., [13 or 4]).


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  • [1] S. I. Adian, The Burnside problem and identities in groups (translated by J. Lennox and J. Wiegold), Springer-Verlag, Berlin and New York, 1979. MR 537580 (80d:20035)
  • [2] L. Fuchs, Infinite abelian groups, Vol. 2, Academic Press, New York, 1970. MR 0255673 (41:333)
  • [3] D. Gildenhuys, W. Herfort and L. Ribes, Profinite Frobenius groups, Arch. Math. (Basel) 33 (1979), 518-528. MR 570487 (81g:20058)
  • [4] S. Grosser and M. Moskowitz, Compactness conditions in topological groups, J. Reine Angew. Math. 246 (1971), 1-40. MR 0284541 (44:1766)
  • [5] S. Grosser, O. Loos and M. Moskowitz, Über Automorphismengruppen lokal-kompakter Gruppen und Derivationen von Lie-Gruppen, Math. Z. 114 (1970), 321-339. MR 0263976 (41:8575)
  • [6] W. Herfort, Compact torsion groups and finite exponent, Arch. Math. (Basel) 33 (1979), 404-410. MR 567358 (81c:22010)
  • [7] G. Hochschild, The structure of Lie groups, Holden-Day, San Francisco, Calif., 1969. MR 0207883 (34:7696)
  • [8] O. H. Kegel and B. A. F. Wehrfritz, Locally finite groups, North-Holland, Amsterdam and London; American Elsevier, New York, 1973. MR 0470081 (57:9848)
  • [9] O. Loos, Symmetric spaces. II, Benjamin, New York, Amsterdam, 1969.
  • [10] J. R. McMullen, Compact torsion groups, Proc. Second Internat. Conf. Theory of Groups, Lecture Notes in Math., Springer, Berlin and New York, 1973. MR 0360845 (50:13292)
  • [11] C. C. Moore, Groups with finite-dimensional irreducible representations, Trans. Amer. Math. Soc. 166 (1972), 401-410. MR 0302817 (46:1960)
  • [12] A. Yu. Olshanskij, An infinite group with its subgroups of prime order, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), 309-321. MR 571100 (82a:20035)
  • [13] T. W. Palmer, Classes of nonabelian noncompact locally compact groups, Rocky Mountain J. Math. 8 (1978), 683-741. MR 513952 (81j:22003)
  • [14] V. P. Platonov, Periodic and compact subgroups of topological groups, Sibirsk. Mat. Ž. 7 (1966), 854-877. (Russian) MR 0199312 (33:7460)
  • [15] L. Ribes, Introduction to profinite groups and Galois cohomology, Queen's Papers in Pure and Appl. Math. vol. 24, Queen's Univ., Kingston, Ont., 1970. MR 0260875 (41:5495)
  • [16] D. J. S. Robinson, Finiteness conditions and generalized soluble groups, part 1, Springer, Berlin and New York, 1972. MR 0332989 (48:11314)
  • [17] J.-P. Serre, Cohomologie Galoisienne, Lecture Notes in Math., vol. 5, Springer, Berlin, 1965. MR 0201444 (34:1328)
  • [18] E. Thoma, Über unitäre Darstellungen abzädhlbarer diskreter Gruppen, Math. Ann. 153 (1964), 111-132. MR 0160118 (28:3332)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0735417-4
Keywords: Compactness conditions, profinite theory, Lie groups, Moore groups
Article copyright: © Copyright 1984 American Mathematical Society

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