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

TL;DR

This paper introduces the generalized wreath product of permutation groups to analyze the schurity problem for S-rings over cyclic groups, providing criteria for schurity and characterizing non-schurian cases.

Contribution

It develops a new theoretical framework using generalized wreath products to determine schurity of S-rings over cyclic groups and characterizes non-schurian structures for specific prime factor counts.

Findings

G is a Schur group if Ω(n) ≤ 3

Criteria for schurity and non-schurity of generalized wreath products

Structure description of non-schurian S-rings when Ω(n)=4

Abstract

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.