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

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 $\mathrm{\Omega }(n)$ of prime factors of the integer $n=\mathrm{\mid }G\mathrm{\mid }$ is at most 3. Moreover, we describe the structure of a non-schurian S-ring over $G$ when $\mathrm{\Omega }(n)=4$. The latter result implies in particular that if $n=p^{3}q$ where $p$ and $q$ are primes, then $G$ is a Schur group.