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


Abstract in English

The generalized wreath product of permutation groups is introduced. By means of it we study the schurity problem for S-rings over a cyclic group $G$ and the automorphism groups of them. Criteria for the schurity and non-schurity of the generalized wreath product of two such S-rings are obtained. As a byproduct of the developed theory we prove that $G$ is a Schur group whenever the total number $Omega(n)$ of prime factors of the integer $n=|G|$ is at most 3. Moreover, we describe the structure of a non-schurian S-ring over $G$ when $Omega(n)=4$. The latter result implies in particular that if $n=p^3q$ where $p$ and $q$ are primes, then $G$ is a Schur group.

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