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

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

2010-12-24

arxiv.org/abs/1012.5393v3

Mathematics

Combinatorics

45 pages, 1 figure; the notation for generalized wreath product of permuatation groups with regular subgroups is introduced; s

Scientific paper

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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