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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

Characterization of cyclic Schur groups


Authors: S. Evdokimov, I. Kovács and I. Ponomarenko
Original publication: Algebra i Analiz, tom 25 (2013), nomer 5.
Journal: St. Petersburg Math. J. 25 (2014), 755-773
MSC (2010): Primary 05E30, 20B25
DOI: https://doi.org/10.1090/S1061-0022-2014-01315-5
Published electronically: July 18, 2014
MathSciNet review: 3184607
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Abstract: A finite group $ G$ is called a Schur group if any Schur ring over $ G$ is associated in a natural way with a subgroup of $ \mathrm {Sym}(G)$ that contains all right translations. It was proved by R. Pöschel (1974) that, given a prime $ p\ge 5$, a $ p$-group is Schur if and only if it is cyclic. We prove that a cyclic group of order $ n$ is Schur if and only if $ n$ belongs to one of the following five families of integers: $ p^k$, $ pq^k$, $ 2pq^k$, $ pqr$, $ 2pqr$ where $ p$, $ q$, $ r$ are distinct primes, and $ k\ge 0$ is an integer.


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Additional Information

S. Evdokimov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences Fontanka 27, St. Petersburg 191023, Russia
Email: evdokim@pdmi.ras.ru

I. Kovács
Affiliation: IAM and FAMNIT, University of Primorska, Muzejski trg 2, SI6000, Koper, Slovenia
Email: istvan.kovacs@upr.si

I. Ponomarenko
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: inp@pdmi.ras.ru

DOI: https://doi.org/10.1090/S1061-0022-2014-01315-5
Keywords: Schur ring, Schur group, permutation group, circulant cyclotomic S-ring, generalized wreath product
Received by editor(s): September 7, 2012
Published electronically: July 18, 2014
Additional Notes: This work was partially supported by the Slovenian–Russian bilateral project, grants nos. BI-RU/10-11-018 and BI-RU/12-13-035. The third author was also supported by the RFFI grant no. 11-01-00760-a
Article copyright: © Copyright 2014 American Mathematical Society

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