相关论文: Separability of Schur rings over abelian groups of…

Let $G$ be a finite group. Denote by $\textrm{Irr}(G)$ the set of all irreducible complex characters of $G.$ Let $\textrm{cd}(G)=\{\chi(1)\;|\;\chi\in \textrm{Irr}(G)\}$ be the set of all irreducible complex character degrees of $G$…

An order is a commutative ring that as an abelian group is finitely generated and free. A commutative ring is reduced if it has no non-zero nilpotent elements. In this paper we use a new tool, namely, the fact that every reduced order has a…

Let $G$ be a finite $p$-group of order $p^{n}$ with $|M(G)|=p^{\frac{n(n-1)}{2}-t},$ where $M(G)$ is the Schur multiplier of $G$. Ya.G. Berkovich, X. Zhou, and G. Ellis have determined the structure of $G$ when $t=0,1,2,3$. In this paper,…

When considering the unit group of $\mathcal{O}_F G$ ($\mathcal{O}_F$ the ring of integers of an abelian number field $F$ and a finite group $G$) certain components in the Wedderburn decomposition of $FG$ cause problems for known generic…

We call a simplicial complex algebraically rigid if its Stanley-Reisner ring admits no nontrivial infinitesimal deformations, and call it inseparable if does not allow any deformation to other simplicial complexes. Algebraically rigid…

Let $X$ be a compact Riemann surface of genus $g\geq 2$. Let $Aut(X)$ be its group of automorphisms and $G\subseteq Aut(X)$ a subgroup. Sharp upper bounds for $|G|$ in terms of $g$ are known if $G$ belongs to certain classes of groups, e.g.…

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.…

We determine the structure of the finite non-solvable groups of order divisible by $3$ all whose maximal subgroups of order divisible by $3$ are supersolvable. Precisely, we demonstrate that if $G$ is a finite non-solvable group satisfying…

For a finite abelian group $G$, let $\beta_{\mathrm{sep}}(G)$ denote its separating Noether number. We determine $\beta_{\mathrm{sep}}(G)$ exactly for every finite abelian group $ G \cong C_{n_1}\oplus \cdots \oplus C_{n_r}$ with $ 1<n_1…

An abelian group $A$ is said to be cancellable if whenever $A \oplus G$ is isomorphic to $A \oplus H$, $G$ is isomorphic to $H$. We show that the index set of cancellable rank 1 torsion-free abelian groups is $\Pi^0_4$ $m$-complete, showing…

If $G$ is an abelian group, we say $S\subset G$ is a set of recurrence if for every probability measure preserving $G$-system $(X,\mu,T)$ and every $D\subset X$ having $\mu(D)>0$, there is a $g\in S$ such that $\mu(D\cap T^{g}D)>0$. We say…

Let $p$ be a prime number and suppose that every maximal subgroup of a finite group is either $p$-nilpotent or has prime index. Such group need not be $p$-solvable, and we study its structure by proving that only one nonabelian simple group…

Let $G$ be a proper subgroup of $\mathbb{Q}$ and $S_G$ be the set of primes $p$ for which $G$ is $p$-divisible. We show that the model-theoretic Grothendieck ring of the ordered abelian group $(G;+,<)$ is a quotient of…

We give a new, effective proof of the separability of cubically convex-cocompact subgroups of special groups. As a consequence, we show that if $G$ is a virtually compact special hyperbolic group, and $Q\leq G$ is a $K$-quasiconvex…

An abelian group is said to be aleph_1-free if all its countable subgroups are free. Our main result is: If R is a ring with R^+ free and |R|<lambda <= 2^{aleph_0}, then there exists an aleph_1-free abelian group G of cardinality lambda…

We study codes with a single check element derived from group rings, namely, checkable codes. The notion of a code-checkable group ring is introduced. Necessary and sufficient conditions for a group ring to be code-checkable are given in…

The power graph and the enhanced power graph of a group $\mathbf G$ are simple graphs with vertex set $G$; two elements of $G$ are adjacent in the power graph if one of them is a power of the other, and they are adjacent in the enhanced…

It has been proved in \cite{ge} for every $p$-group of order $p^n$, $|\mathcal{M}(G)|=p^{\f{1}{2}n(n-1)-t(G)}$, where $t(G)\geq 0$. In \cite{be, el, zh}, the structure of $G$ has been characterized for $t(G)=0,1,2,3$ by several authors.…

We prove that if $R$ is a ring that is object unital and strongly graded by a groupoid $\Gamma$, and if $\Delta$ is a wide subgroupoid of $\Gamma$, then $R/R_\Delta$ is separable if and only if, for each $e \in \Gamma_0$, there exist $f \in…

Let $G$ be a nonabelian group and $n$ a natural number. We say that $G$ has a strict $n$-split decomposition if it can be partitioned as the disjoint union of an abelian subgroup $A$ and $n$ nonempty subsets $B_1, B_2, \ldots, B_n$, such…