Related papers: Units of twisted group rings and their correlation…

Similarly to how the classical group ring isomorphism problem asks, for a commutative ring $R$, which information about a finite group $G$ is encoded in the group ring $RG$, the twisted group ring isomorphism problem asks which information…

We continue our investigation of a variation of the group ring isomorphism problem for twisted group algebras. Contrary to previous work, we include cohomology classes which do not contain any cocycle of finite order. This allows us to…

Let $RG$ be the gruop ring of the group $G$ over ring $R$ and $\mathscr{U}(RG)$ be its unit group. Finding the structure of the unit group of a finite group ring is an old topic in ring theory. In, G. Tang et al: Unit Groups of Group…

During the past three decades fundamental progress has been made on constructing large torsion-free subgroups (i.e. subgroups of finite index) of the unit group $\U (\Z G)$ of the integral group ring $\Z G$ of a finite group $G$. These…

Let $R$ be a finite commutative ring with identity and $U(R)$ be its group of units. In 2005, El-Kassar and Chehade presented a ring structure for $U(R)$ and as a consequence they generalized this group of units to the generalized group of…

Let U be the group of units of an infinite twisted group algebra K_\lambda G over a field K. We describe the maximal FC-subgroup of U and give a characterization of U with finitely conjugacy classes. In the case of group algebras we obtain…

The Subgroup Isomorphism Problem for Integral Group Rings asks for which finite groups U it is true that if U is isomorphic to a subgroup of V(ZG), the group of normalized units of the integral group ring of the finite group G, it must be…

The twisted group ring isomorphism problem (TGRIP) is a variation of the classical group ring isomorphism problem. It asks whether the ring structure of the twisted group ring determines the group up to isomorphism. In this article, we…

For a finite group $G$ and $U: = U(\mathbb{Z}G)$, the group of units of the integral group ring of $G$, we study the implications of the structure of $G$ on the abelianization $U/U'$ of $U$. We pose questions on the connections between the…

We propose and study a variation of the classical isomorphism problem for group rings in the context of projective representations. We formulate several weaker conditions following from our notion and give all logical connections between…

Let $G$ be a finite group, $\Z G$ the integral group ring of $G$ and $\U(\Z G)$ the group of units of $\Z G$. The Congruence Subgroup Problem for $\U(\Z G)$ is the problem of deciding if every subgroup of finite index of $\U(\Z G)$ contains…

In the 1940's Graham Higman initiated the study of finite subgroups of the unit group of an integral group ring. Since then many fascinating aspects of this structure have been discovered. Major questions such as the Isomorphism Problem and…

In this article, we explore the problem of determining isomorphisms between the twisted complex group algebras of finite $p$-groups. This problem bears similarity to the classical group algebra isomorphism problem and has been recently…

We give a unified description of twisted forms of classical reductive groups schemes. Such group schemes are constructed from algebraic objects of finite rank, excluding some exceptions of small rank. These objects, augmented odd form…

An isomorphism between the group ring of a finite group and a ring of certain block diagonal matrices is established. The group ring $RG$ of a finite group $G$ is isomorphic to the set of {\em group ring matrices} over $R$. It is shown that…

We study $(\sigma,\tau)$-derivations of a group ring $RG$ where $G$ is a group with center having finite index in $G$ and $R$ is a semiprime ring with $1$ such that either $R$ has no torsion elements or that if $R$ has $p$-torsion elements,…

We introduce a directed graph related to a group $G$, which we call the N-prime graph $\Gamma_{\rm{N}}(G)$ of $G$ and which is a refinement of the classical Gruenberg-Kegel graph. The vertices of $\Gamma_{\rm{N}}(G)$ are the primes $p$ such…

We consider a locally compact Hausdorff groupoid $G$, and twist by a more general locally compact Hausdorff abelian group $\Gamma$ rather than the complex unit circle $\mathbb{T}$. We investigate the construction of $C^*$-algebras in…

This study introduces a new unified structural framework for orbifold sigma models that incorporates twisted sectors, singularities, and smooth regions into a single algebraic object. Traditional approaches to orbifold theories often treat…

We introduce separability properties corresponding to generalized versions of the conjugacy, twisted conjugacy, Brinkmann and Brinkmann's conjugacy problems and how they relate when finite and cyclic extensions of groups are taken. In…