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Extension of a theorem of Baayen and Helmberg on monothetic groups


Author: D. L. Armacost
Journal: Proc. Amer. Math. Soc. 96 (1986), 502-504
MSC: Primary 22A05
DOI: https://doi.org/10.1090/S0002-9939-1986-0822449-8
MathSciNet review: 822449
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Abstract: Let $ G$ and $ K$ be compact monothetic groups and let $ \phi $ be a continuous homomorphism from $ G$ onto $ K$. If $ k$ is a generator of $ K$, must there exist a generator $ g$ of $ G$ such that $ \phi \left( g \right) = k?$? A useful theorem of Baayen and Helmberg provides an affirmative answer if $ K$ is the circle $ T$. We show that the answer remains affirmative as long as $ K$ is metrizable. We also provide an example to show that the answer may be negative for nonmetrizable $ K$.


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DOI: https://doi.org/10.1090/S0002-9939-1986-0822449-8
Keywords: Monothetic, generator
Article copyright: © Copyright 1986 American Mathematical Society

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