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Schurity of S-rings over a cyclic group and generalized wreath product of permutation groups. (English. Russian original) Zbl 1278.20002
St. Petersbg. Math. J. 24, No. 3, 431-460 (2013); translation from Algebra Anal. 24, No. 3, 84-127 (2012).
Summary: With the help of the generalized wreath product of permutation groups introduced in the paper, the automorphism group of an S-ring over a finite cyclic group \(G\) is studied. Criteria for the generalized wreath product of two such 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 S-ring over it is Schurian) whenever the total number \(\Omega(n)\) of prime factors of the integer \(n=|G|\) does not exceed 3. Moreover, the structure of a non-Schurian 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.

20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20E22 Extensions, wreath products, and other compositions of groups
16S34 Group rings
05E30 Association schemes, strongly regular graphs
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
Full Text: DOI
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