Related papers: On general linear groups over exchange rings

The current paper is an addition to the previous paper by author, where the overgroup lattice of the elementary subsystem subgroup $E(\Delta,R)$ of the Chevalley group $G(\Phi,R)$ for a large enough root subsystem $\Delta$ was studied. Now…

Let $D$ be a division ring with infinite center, $n$ be a positive integer and $w(x_1,x_2,\cdots, x_m)=1$ be a generalized group identity over the general linear group $\GL_n(D)$. The aim of this paper is to prove that every subnormal…

Let $R$ be a commutative ring with unity. Consider the twisted Chevalley group $G_{\pi, \sigma} (\Phi, R)$ of type $\phi$ over $R$ and its elementary subgroup $E'_{\pi, \sigma} (\Phi, R)$. This paper investigates the normalizers of…

In this paper we describe all groups that are first-order (elementarily) equivalent to the classical matrix groups such as $GL_n(F), SL_n(F)$ and $T_n(F)$ over a field $F$ provided $n \geq 3$.

Let $G$ be a transitive normal subgroup of a permutation group $A$ of finite degree $n$. The factor group $A/G$ can be considered as a certain Galois group and one would like to bound its size. One of the results of the paper is that $|A/G|…

Let $(M, q)$ be a quadratic projective module of an odd rank over an commutative ring, where the form $q$ is semiregular, with global Witt index of at least $2$, and with $\mathrm{rk}(M) \ge 7$. We prove standard commutator formulae and…

Let $X_n = \{1,2,\dots,n\}$ be a finite set $(n\geq 2)$ and $T_n$ the full transformation semigroup on $X_n$. For a positive integer $l\leq n-1$, we define $$T_n(l) = \{\alpha\in T_n \colon \forall x,y\in X_n,\, |x-y| = l \;\Rightarrow\;…

Let G be a finitely generated group and M_n(G) the number of its normal subgroup subgroups of index at most n. For linear groups G we show that M_n(G) can grow polynomially in n only if the semisimple part of the Zariski closure of G has…

The normal covering number $\gamma(G)$ of a finite, non-cyclic group $G$ is the minimum number of proper subgroups such that each element of $G$ lies in some conjugate of one of these subgroups. We find lower bounds linear in $n$ for…

This article studies the equation $[A,B]^k = {\rm Id}_n$ for matrices over $\mathbb{C}$, characterizing the pairs $(k,n)$ for which solutions exist via a classical result of Lam and Leung on sums of roots of unity. The problem is next…

In this paper we prove that the Hermite ring conjecture holds for valuation rings $V$, and the special liner group $SL_n(V[x])$ coincides with the group generated by elementary matrices $E_n(V[x])$ for $n\geq3$. For any arithmetical ring…

We generalise a key result of one-relator group theory, namely Magnus's Freiheitssatz, to partially commutative groups, under sufficiently strong conditions on the relator. The main theorem shows that under our conditions, on an element $r$…

It is known that the notion of a transitive subgroup of a permutation group $G$ extends naturally to subsets of $G$. We consider subsets of the general linear group $\operatorname{GL}(n,q)$ acting transitively on flag-like structures, which…

In this study, it is proven that the universal equivalence of general linear groups (admitting the inverse-transpose automorphism) of orders greater than $2$, over local, not necessarily commutative rings with $1/2$, is equivalent to the…

In this paper, we study almost subnormal subgroups of the general linear group $\GL_n(D)$ of degree $n\ge 1$ over a division ring $D$ that satisfy a generalized power central group identity.

For a commutative ring R with many units, we describe the kernel of H_3(inc): H_3(GL_2(R), Z) --> H_3(GL_3(R), Z). Moreover we show that the elements of this kernel are of order at most two. As an application we study the indecomposable…

Let $r$ and $n$ be positive integers, let $G_n$ be the complex reflection group of $n \times n$ monomial matrices whose entries are $r^{\textrm{th}}$ roots of unity and let $0 \leq k \leq n$ be an integer. Recently, Haglund, Rhoades and…

We show that given a finitely generated standard-graded algebra of dimension $d$ over an infinite field, its graded Noether normalizations obey a certain kind of `generic exchange', allowing one to pass between any two of them in at most…

Let $G=C_{p^n}$ be a finite cyclic p-group, and let $Hol(G)$ denote its holomorph. In this work, we find and characterize the regular subgroups of $Hol(G)$ that are mutually normalizing each other in the permutation group $Sym(G)$. We…

In this short note, we prove that there is a well behaved notion of normal hull for smooth algebraic group schemes over a field and that the commutator group $(G,H)$ is well defined for $H\subset G$ smooth, even when both of them are not…