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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.
The recently developed theory of Schur rings over a finite cyclic group is generalized to Schur rings over a ring R being a product of Galois rings of coprime characteristics. It is proved that if the characteristic of R is odd, then as in the cyclic
A subset $B$ of an Abelian group $G$ is called a difference basis of $G$ if each element $gin G$ can be written as the difference $g=a-b$ of some elements $a,bin B$. The smallest cardinality $|B|$ of a difference basis $Bsubset G$ is called the diffe
A commutative ring R has finite rank r, if each ideal of R is generated at most by r elements. A commutative ring R has the r-generator property, if each finitely generated ideal of R can be generated by r elements. Such rings are closely related to
We define an excedance number for the multi-colored permutation group, i.e. the wreath product of Z_{r_1} x ... x Z_{r_k} with S_n, and calculate its multi-distribution with some natural parameters. We also compute the multi-distribution of the par
Powering the adjacency matrix of an expander graph results in a better expander of higher degree. In this paper we seek an analogue operation for high-dimensional expanders. We show that the naive approach to powering does not preserve high-dimension