Related papers: On Schur Rings Over Infinite Groups
A finite group $G$ is called a Schur group if every Schur ring over $G$ is schurian, i.e. associated in a natural way with a subgroup of the symmetric group $Sym(G)$ that contains all right translations of $G$. The list of all possible…
Let $G$ be a finite group. If $\Gamma$ is a permutation group with $G_{right}\leq\Gamma\leq Sym(G)$ and $\mathcal{S}$ is the set of orbits of the stabilizer of the identity $e=e_{G}$ in $\Gamma$, then the $\mathbb{Z}$-submodule…
In this paper, in the first we give definitions of some classes of division rings which strictly contain the class of centrally finite division rings. One of our main purpose is to construct non-trivial examples of rings of new defined…
The paper concerns the automorphism groups of Cayley graphs over cyclic groups which have a rational spectrum (rational circulant graphs for short). With the aid of the techniques of Schur rings it is shown that the problem is equivalent to…
In 1993, Muzychuk \cite{muzychuk} showed that the rational Schur rings over a cyclic group $Z_n$ are in one-to-one correspondence with sublattices of the divisor lattice of $n$, or equivalently, with sublattices of the lattice of subgroups…
A finite group $G$ is called a Schur group if every Schur ring over $G$ is schurian, i.e. associated in a natural way with a subgroup of the symmetric group $Sym(G)$ that contains all right translations of $G$. The list of all possible…
Recently, it was proved that every commutative $p$-Schur ring over a group of order $p^3$ is Schurian. In this article, we consider the Schurity problem of non-commutative $p$-Schur rings over groups of order $p^3$. In particular, it is…
An $S$-ring (a Schur ring) is said to be separable with respect to a class of groups $\mathcal{K}$ if every algebraic isomorphism from the $S$-ring in question to an $S$-ring over a group from $\mathcal{K}$ is induced by a combinatorial…
An $S$-ring (a Schur ring) is said to be separable with respect to a class of groups $\mathcal{K}$ if every algebraic isomorphism from the $S$-ring in question to an $S$-ring over a group from $\mathcal{K}$ is induced by a combinatorial…
In this paper we shall be looking at several results relating Schur rings to sufficient conditions for a graph to be a graphical regular representation (GRR) of a finite group, and then applying these specifically in the case of certain…
In \cite[Problem 72]{Fuchs60} Fuchs posed the problem of characterizing the groups which are the groups of units of commutative rings. In the following years, some partial answers have been given to this question in particular cases. In a…
In this paper, we shall show that the only primitive Schur ring over a semi dihedral group is the trivial one and every semi dihedral subgroup is Burnside group, that is a primitive group containing a regular subgroup isomorphic to the semi…
A Schur ring over a finite group is said to be decomposable if it is the generalized wreath product of Schur rings over smaller groups. In this paper we establish a sufficient condition for a decomposable Schur ring over the direct product…
The Schur Theorem says that if $G$ is a group whose center $Z(G)$ has finite index $n$, then the order of the derived group $G'$ is finite and bounded by a number depending only on $n$. In the present paper we show that if $G$ is a finite…
We introduce the notion of a Schur-finite element in a $\lambda$-ring.
A finite group is said to be weakly separable if every algebraic isomorphism between two $S$-rings over this group is induced by a combinatorial isomorphism. In the paper we prove that every abelian weakly separable group belongs to one of…
We make a list of finite simple groups whose group rings over a given field are serial.
For any finite group $G$ and a positive integer $m$, we define andstudy a Schur ring over the direct power $G^m$, which gives an algebraic interpretation of the partition of $G^m$ obtained by the $m$-dimensional Weisfeiler-Leman algorithm.…
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…
We study commutative Schur rings over the symplectic groups Sp$(n,2)$ containing the class $\mathcal C$ of symplectic transvections. We find the possible partitions of $\mathcal C$ determined by the Schur ring. We show how this restricts…