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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

   

 

Schurity of $ S$-rings over a cyclic group and generalized wreath product of permutation groups


Authors: S. A. Evdokimov and I. N. Ponomarenko
Translated by: the authors
Original publication: Algebra i Analiz, tom 24 (2012), nomer 3.
Journal: St. Petersburg Math. J. 24 (2013), 431-460
MSC (2010): Primary 20E22; Secondary 20J05
Published electronically: March 21, 2013
MathSciNet review: 3014128
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Abstract: With the help of the generalized wreath product of permutation groups introduced in the paper, the automorphism group of an $ \textup {S}$-ring over a finite cyclic group $ G$ is studied. Criteria for the generalized wreath product of two such $ \textup {S}$-rings to be Schurian or non-Schurian are proved. As a byproduct, it is shown that the group $ G$ is a Schur one (i.e., any $ \textup {S}$-ring over it is Schurian) whenever the total number $ \Omega (n)$ of prime factors of the integer $ n=\vert G\vert$ does not exceed $ 3$. Moreover, the structure of a non-Schurian $ \textup {S}$-ring over $ G$ is described in the case where $ \Omega (n)=4$. In particular, the last result implies that if $ n=p^3q$, where $ p$ and $ q$ are primes, then $ G$ is a Schur group.


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Additional Information

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

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

DOI: http://dx.doi.org/10.1090/S1061-0022-2013-01246-5
Keywords: Shurian ring, generalized wreath product, permutation group
Received by editor(s): April 7, 2011
Published electronically: March 21, 2013
Additional Notes: Partially supported by the Slovenian–Russian bilateral project (grant nos. BI-RU/10-11-018 and BI-RU/12-13-035). The second author was also supported by RFBR (grant no. 11-01-00760-a)
Article copyright: © Copyright 2013 American Mathematical Society