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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
DOI: https://doi.org/10.1090/S0002-9939-1982-0652437-2
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?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0652437-2
Keywords: Endomorphism ring, topological group, abelian group, representable ring
Article copyright: © Copyright 1982 American Mathematical Society

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