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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Discontinuity of multiplication and left translations in $\beta G$
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by Yevhen Zelenyuk PDF
Proc. Amer. Math. Soc. 143 (2015), 877-884 Request permission


The operation of a discrete group $G$ naturally extends to the Stone-Čech compactification $\beta G$ of $G$ so that for each $a\in G$, the left translation $\beta G\ni x\mapsto ax\in \beta G$ is continuous, and for each $q\in \beta G$, the right translation $\beta G\ni x\mapsto xq\in \beta G$ is continuous. We show that for every Abelian group $G$ with finitely many elements of order 2 such that $|G|$ is not Ulam-measurable and for every $p,q\in G^*=\beta G\setminus G$, the multiplication $\beta G\times \beta G\ni (x,y)\mapsto xy\in \beta G$ is discontinuous at $(p,q)$. We also show that it is consistent with ZFC, the system of usual axioms of set theory, that for every Abelian group $G$ and for every $p,q\in G^*$, the left translation $G^*\ni x\mapsto px\in G^*$ is discontinuous at $q$.
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Additional Information
  • Yevhen Zelenyuk
  • Affiliation: School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa
  • Email:
  • Received by editor(s): February 19, 2013
  • Received by editor(s) in revised form: May 18, 2013
  • Published electronically: October 6, 2014
  • Additional Notes: The author was supported by NRF grant IFR2011033100072.
  • Communicated by: Mirna Džamonja
  • © Copyright 2014 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 143 (2015), 877-884
  • MSC (2010): Primary 03E35, 22A15; Secondary 22A05, 54D35
  • DOI:
  • MathSciNet review: 3283674