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On endomorphisms of abelian topological groups

Authors: Eli Katz and Sidney A. Morris
Journal: Proc. Amer. Math. Soc. 85 (1982), 181-183
MSC: Primary 22B05; Secondary 16A65
MathSciNet review: 652437
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Abstract: A family $ \Phi $ of continuous endomorphisms of a topological group $ G$ is said to be small if for every subgroup $ H$ of $ G$ of cardinality $ {\text{card}}(H) < {\text{card}}(G)$ there exists an element $ g \in G$ such that $ \Phi g \cap H = \emptyset$. M. I. Kabenjuk [5] proved that if $ G$ is a compact connected Hausdorff abelian group of countable weight then every countable family $ \Phi $ of nontrivial endomorphisms of $ G$ is small. He asked if "compact" can be replaced by "complete". In this note the answer is given in the negative, but it is shown that "compact" can be replaced by "locally compact".

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

  • [1] L. Fuchs, Infinite abelian groups. Vol. II, Academic Press, New York, 1973. MR 0349869 (50:2362)
  • [2] David C. Hunt and Sidney A. Morris, Free subgroups of topological groups, Proc. Second Internat. Conf. Theory of Groups, Canberra, Lecture Notes in Math., vol. 372, Springer-Verlag, Berlin and New York, 1974, pp. 377-387. MR 0352317 (50:4804)
  • [3] M. I. Kabenjuk, Finite groups of automorphisms of topological groups, Algebra and Logic 13 (1974), 291-299. (Russian) MR 0369491 (51:5724)
  • [4] -, Groups of automorphisms of topological groups, Algebra Symposium Proceedings. I, Gomel, 1975. (Russian)
  • [5] -, Some topologies on abelian groups related to the ring of its endomorphisms, Colloquia Mathematica, Societatis János Bolyai 23 Topology (Budapest), 1978, pp. 705-712. MR 588818 (82d:16030)
  • [6] John Mack, Sidney A. Morris and Edward T. Ordman, Free topological groups and the projective dimension of a locally compact abelian group, Proc. Amer. Math. Soc. 40 (1973), 303-308. MR 0320216 (47:8755)
  • [7] Sidney A. Morris, Varieties of topological groups and left adjoint functions, J. Austral. Math. Soc. 16 (1973), 220-227. MR 0333059 (48:11384)
  • [8] -, Pontryagin duality and the structure of locally compact abelian groups, London Math. Soc. Lecture Notes Series, no. 29, Cambridge Univ. Press, London, 1977. MR 0442141 (56:529)
  • [9] -, Varieties of topological groups, Bull. Austral. Math. Soc. 1 (1969), 145-160. MR 0259010 (41:3655a)
  • [10] R. J. Wille, The existence of a topological group with automorphism group $ {C_7}$, Quart J. Math. Oxford Ser. (2) 18 (1967), 53-57. MR 0228617 (37:4197)

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Keywords: Endomorphism ring, topological group, abelian group, representable ring
Article copyright: © Copyright 1982 American Mathematical Society

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