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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.

MSC:
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
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