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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality 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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Large groups and their periodic quotients
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by A. Yu. Olshanskii and D. V. Osin PDF
Proc. Amer. Math. Soc. 136 (2008), 753-759 Request permission

Abstract:

We first give a short group theoretic proof of the following result of Lackenby. If $G$ is a large group, $H$ is a finite index subgroup of $G$ admitting an epimorphism onto a non-cyclic free group, and $g_1, \ldots , g_k$ are elements of $H$, then the quotient of $G$ by the normal subgroup generated by $g_1^n, \ldots , g_k^n$ is large for all but finitely many $n\in \mathbb Z$. In the second part of this note we use similar methods to show that for every infinite sequence of primes $(p_1, p_2, \ldots )$, there exists an infinite finitely generated periodic group $Q$ with descending normal series $Q=Q_0\rhd Q_1\rhd \ldots$, such that $\bigcap _i Q_i=\{ 1\}$ and $Q_{i-1}/Q_i$ is either trivial or abelian of exponent $p_i$.
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Additional Information
  • A. Yu. Olshanskii
  • Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240; and Department of Mathematics, Moscow State University, Moscow, 119899, Russia
  • MR Author ID: 196218
  • Email: alexander.olshanskiy@vanderbilt.edu
  • D. V. Osin
  • Affiliation: Department of Mathematics, The City College of New York, New York, New York 10031
  • MR Author ID: 649248
  • Email: denis.osin@gmail.com
  • Received by editor(s): April 19, 2006
  • Published electronically: November 26, 2007
  • Additional Notes: The first author was supported in part by the NSF grants DMS 0245600 and DMS 0455881.
    The second author was supported in part by NSF grant DMS 0605093. Both authors were supported in part by the Russian Fund for Basic Research grant 05-01-00895.
  • Communicated by: Jonathan I. Hall
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 136 (2008), 753-759
  • MSC (2000): Primary 20F50, 20F05, 20E26
  • DOI: https://doi.org/10.1090/S0002-9939-07-09150-2
  • MathSciNet review: 2361846