Related papers: On Schur Rings Over Infinite Groups

Schur rings over the infinite dihedral group $\mathcal{Z}\rtimes\mathcal{Z}_2$ are studied according to properties of Schur rings over infinite groups and the classification of Schur rings over infinite cyclic groups. Schur rings over…

We generalize the idea of a Schur ring of a group to the category of semigroups. Fundamental results of Schur rings over groups are shown to be true for Schur rings over semigroups. Examples where Schur rings differ between the two…

For the direct product $\cZ\times \cZ_3$ of infinite cyclic group $\cZ$ and a cyclic group $\cZ_3$ of order $3$, the schur rings over it are classified. In particular, all the schur rings are proved to be traditional.

In the paper, we develop further the properties of Schur rings over infinite groups, with particular emphasis on the virtually cyclic group $\Z\times\Z_p$, where $p$ is a prime. We provide structure theorems for primitive sets in these…

Schur rings are a type of subrings of group rings afforded by a partition of the underlined group. In this paper, Schur rings over free abelian group of rank two are classified under the assumption that one of the direct factor is a union…

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…

Any Schur ring is uniquely determined by a partition of the elements of the group. In this paper we present a general technique for enumerating Schur rings over cyclic groups using traditional Schur rings. We also survey recent efforts to…

An $S$-ring (Schur ring) is called central if it is contained in the center of the group ring. We introduce the notion of a generalized Schur group, i.e. such finite group that all central $S$-rings over this group are schurian. It…

A finite group $G$ is called a Schur group, if any Schur ring over $G$ is associated in a natural way with a subgroup of $Sym(G)$ that contains all right translations. Recently, the authors have completely identified the cyclic Schur…

In this paper, we explore the nature of central idempotents of Schur rings over finite groups. We introduce the concept of a lattice Schur ring and explore properties of these kinds of Schur rings. In particular, the primitive, central…

Any Schur ring is uniquely determined by a partition of the elements of the group. An open question in the study of Schur rings is determining which partitions of the group induce a Schur ring. Although a structure theorem is available for…

A finite group $G$ is called a Schur group if every $S$-ring over $G$ is schurian, i.e. associated in a natural way with a subgroup of $\sym(G)$ that contains all right translations. We prove that every nonabelian nilpotent Schur group…

A finite group $G$ is called a Schur group, if any Schur ring over $G$ is the transitivity module of a permutation group on the set $G$ containing the regular subgroup of all right translations. It was proved by R. P\"oschel (1974) that…

A finite group $G$ is called a Schur group if any $S$-ring over $G$ is associated in a natural way with a subgroup of $Sym(G)$ that contains all right translations. We prove that the groups $\mathbb{Z}_3\times \mathbb{Z}_{3^n}$, where…

The famous Burnside-Schur theorem states that every primitive finite permutation group containing a regular cyclic subgroup is either 2-transitive or isomorphic to a subgroup of a 1-dimensional affine group of prime degree. It is known that…

A Schur ring (S-ring) over a group $G$ is called separable if every of its similaritities is induced by isomorphism. We establish a criterion for an S-ring to be separable in the case when the group $G$ is cyclic. Using this criterion, we…

In this paper, we continue the enumeration of Schur rings over cyclic groups. Cyclic groups of semiprime order $pq$, where $p$ and $q$ are distinct primes, are considered. Additionally, cyclic groups of order $4p$ are considered.

A finite group $G$ is called a Schur group, if any Schur ring over $G$ is the transitivity module of a point stabilizer in a subgroup of $\sym(G)$ that contains all right translations. We complete a classification of abelian $2$-groups by…

A finite group $G$ is called a Schur group if every $S$-ring over $G$ is schurian, i.e. associated in a natural way with a subgroup of $Sym(G)$ that contains all right translations. One of the crucial questions in the $S$-ring theory is the…

A finite group $G$ is called a Schur group if any Schur ring over $G$ is associated in a natural way with a subgroup of $Sym(G)$ that contains all right translations. It is proved that the group $C_3\times C_3\times C_p$ is Schur for any…