Related papers: Narain Gupta's three normal subgroup problem and g…

A topological group $G$ is said to have no small subgroup (resp. no small normal subgroup) if it admits an open neighbourhood of the identity containing no non-trivial subgroup (resp. normal subgroup) of $G$. These properties are usually…

We review some of our recent results concerning the relationship between the Real-Space Renormalization Group method and Quantum Groups. We show this relation by applying real-space RG methods to study two quantum group invariant…

This article focuses on the study of cut groups, i.e., the groups which have only trivial central units in their integral group ring. We provide state of art for cut groups. The results are compiled in a systematic manner and have also been…

A finite group $G$ admits a normal $2$-covering if there exist two proper subgroups $H$ and $K$ with $G=\bigcup_{g\in G}H^g\cup\bigcup_{g\in G}K^g$. For determining inductively the finite groups admitting a normal $2$-covering, it is…

One of the most fundamental problems in the theory of finite- dimensional Hopf algebras is their classification over an algebraically closed field k of characteristic 0. This problem is extremely difficult, hence people restrict it to…

Let G be a group, and H a G-group defined by an imbedding map $G\rightarrow H$; in [12] we have defined a topology on a subset of normal subgroups of $H$, the so-called prime ideals. In this work, we generalize this topology to other…

Let G be a finite group. A subgroup M of G is said to be an NR-subgroup if, whenever K is normal in M, then K^G\cap M=K, where K^G is the normal closure of K in G. Using the Classification of Finite Simple Groups, we prove that if every…

We formulate and answer Gorenstein projective, flat, and injective analogues of a classical projectivity question for group rings under some mild additional assumptions. Although the original question, that was proposed by Jang-Hyun Jo in…

Consider an equidimensional faithful conical action of an algebraic torus $T$ on an affine normal conical variety $X$ over an algebraically closed field of characteristic zero. Then there exists a finite normal subgroup $N$ of $T$ such that…

Let $G$ be the fundamental group of a three-manifold. By piecing together many known facts about three manifold groups, we establish two properties of the group ring $\mathbb{C}G$. We show that if $G$ has rational cohomological dimension…

In this paper we propose right-angled Artin groups as a platform for secret sharing schemes based on the efficiency (linear time) of the word problem. Inspired by previous work of Grigoriev-Shpilrain in the context of graphs, we define two…

Let $\omega$ be a Morse form on a manifold $M$. Let $p:\hat M\to M$ be a regular covering with structure group $G$, such that $p^*([\omega])=0$. Let $\xi:G\to\mathbf{R}$ be the corresponding period homomorphism. Denote by ${\hat…

In this paper we consider the Fitting subgroup $F(G)$ of a finite group $G$ and its generalizations: the quasinilpotent radical $F^*(G)$ and the generalized Fitting subgroup $\tilde{F}(G)$ defined by $\tilde{F}(G)\supseteq \Phi(G)$ and…

Let G be a semisimple Lie group with associated symmetric space D, and let Gamma subset G be a cocompact arithmetic group. Let L be a lattice inside a Z Gamma-module arising from a rational finite-dimensional complex representation of G.…

We introduce two minimality properties of subgroups in topological groups. A subgroup $H$ is a key subgroup (co-key subgroup) of a topological group $G$ if there is no strictly coarser Hausdorff group topology on $G$ which induces on $H$…

Using an idea due to R.Thomason, we define a "homology theory" on the category of rings which satisfies excision, exactness, homotopy (in the algebraic sense) and periodicity of order 4. For regular noetherian rings, we find P. Balmer's…

Let $\frak{F}$ be a class of finite groups. A subgroup $H$ of a finite group $G$ is said to be $\mathfrak{F_{\mathrm s}}$-quasinormal in $G$ if there exists a normal subgroup $T$ of $G$ such that $HT$ is $s$-permutable in $G$ and $(H\cap…

A perturbative renormalization group is formulated for the study of Hamiltonian light-front field theory near a critical Gaussian fixed point. The only light-front renormalization group transformations found that can be approximated by…

Suppose that $(T,\star)$ is a groupoid with a left identity such that each element $a\in T$ has a left inverse. Then $T$ is called a \textit{gyrogroup} if and only if $(i)$ there exists a function $gyr:T\times T\longrightarrow Aut(T)$ such…

It is shown that a nontrivial normal subgroup $N$ of a group $G$ is a free factor of the $N$'s normal closure in the $G$'s free product with arbitrary nontrivial groups.