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

Abstract:

With the help of the generalized wreath product of permutation groups introduced in the paper, the automorphism group of an $\mathrm {S}$-ring over a finite cyclic group $G$ is studied. Criteria for the generalized wreath product of two such $\mathrm {S}$-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 $\mathrm {S}$-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 $\mathrm {S}$-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.