Related papers: On the Unit Conjecture for Supersoluble Group Ring…

Let $G$ be a finite abelian group and $S$ a sequence with elements of $G$. Let $|S|$ denote the length of $S$ and $\mathrm{supp}(S)$ the set of all the distinct terms in $S$. For an integer $k$ with $k\in [1, |S|]$, let $\Sigma_{k}(S)…

Let $K$ be a subgroup of a finite group $G$. The probability that an element of $G$ commutes with an element of $K$ is denoted by $Pr(K,G)$. Assume that $Pr(K,G)\geq\epsilon$ for some fixed $\epsilon>0$. We show that there is a normal…

We study the impact of certain identities and probabilistic identities on the structure of finite groups. More specifically, let $w$ be a nontrivial word in $d$ distinct variables and let $G$ be a finite group for which the word map…

In the current article we study structure of a Chevalley group $G(R)$ over a commutative ring $R$. We generalize and improve the following results: (1) standard, relative, and multi-relative commutator formulas; (2) nilpotent structure of…

We say that a class of finite structures for a finite first-order signature is $r$-compressible if each structure $G$ in the class has a first-order description of size at most $O(r(|G|))$. We show that the class of finite simple groups is…

We show that there is an order-preserving embedding of the additive group of rational numbers $\mathbb{Q}$ into a 2-generator group $G$. The group $G$ can be chosen to be a solvable group $G$ of length 3, which is a minimal result in the…

Using the recent advancements in the structure of algebraic groups over imperfect fields, we propose a generalization of Serre's Conjecture I and of results that revolve around it. In particular, we prove that the first Galois cohomology…

Let $G$ be a profinite group. We prove that the commutator subgroup $G'$ is finite-by-procyclic if and only if the set of all commutators of $G$ is contained in a union of countably many procyclic subgroups.

The reducible K\"ahler coset space G/S\otimesU(1)^k is discussed in a geometrical approach. We derive the formula which expresses the Riemann curvature of the reducible K\"ahler coset space in terms of its Killing vectors. The formula…

For a finite group $G$ denote by $N(G)$ the set of conjugesy class sizes of $G$. We show that every finite group $G$ with the property $N(G)=N(Alt_n), n>4$ or $N(G)=N(Sym_n), n>22$ is non-solvable.

We prove that the projective unit group $\mathrm{PGL}(R)$, i.e., the quotient of the unit group $\mathrm{GL}(R)$ modulo its center, of any non-discrete irreducible, continuous ring $R$ is simple. Moreover, we show that $\mathrm{GL}(R)$ has…

This article determines the structure of the group ring $\mathbb{Z}_nG$, where $G$ is a finite group and $\mathbb{Z}_n$ is the ring of integers modulo $n$, such that $n$ is relatively prime to the order of $G$. The decomposition of…

This paper provides an overview to three recent papers on the bottom up approach to GUTs in F-theory. We assume only a minimal familiarity with string theory and phenomenology. After explaining the potential for predictive string…

Let p be a prime, K a field of characteristic p, G a locally finite p-group, KG the group algebra, and V the group of the units of KG with augmentation 1. The anti-automorphism g\mapsto g^{-1} of G extends linearly to KG; this extension…

If G is a simple non-compact Lie group, with K its maximal compact subgroup, such that K contains a one-dimensional center C, then the coset space G/K is an Hermitian symmetric non-compact space. SL(2,R)/U(1) is the simplest example of such…

An unrefinable chain of a finite group $G$ is a chain of subgroups $G = G_0 > G_1 > \cdots > G_t = 1$, where each $G_i$ is a maximal subgroup of $G_{i-1}$. The length (respectively, depth) of $G$ is the maximal (respectively, minimal)…

A subgroup $H$ of a finite group $G$ is called submodular in $G$, if we can connect $H$ with $G$ by a chain of subgroups, each of which is modular (in the sense of Kurosh) in the next. If a group $G$ is supersoluble and every Sylow subgroup…

Let $N$ and $H$ be groups, and let $G$ be an extension of $H$ by $N$. In this article we describe the structure of the complex group ring of $G$ in terms of data associated with $N$ and $H$. In particular, we present conditions on the…

Let $k$ be an algebraically closed field of characteristic $p>0$ and $G$ a finite group. We provide a description of the torsion subgroup $TT(G)$ of the finitely generated abelian group $T(G)$ of endo-trivial $kG$-modules when $p=2$ and $G$…

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…