Related papers: On general linear groups over exchange rings

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…

For a non-cyclic finite group $G$, let $\gamma(G)$ denote the smallest number of conjugacy classes of proper subgroups of $G$ needed to cover $G$. Bubboloni, Praeger and Spiga, motivated by questions in number theory, have recently…

In this article, we give explicit proofs of certain commutator relations among the elementary generators of the elementary orthogonal group $EO_A(Q\perp H(P))$, where $A$ is a commutative ring, $Q$ is a non-singular quadratic $A$-space and…

In this paper, we give a characterization of the action of any abelian subgroup G of GL(n, C) on C^n. We prove that any orbit of G is regular with order m<=2n. Moreover, we give a method to determine this order. In the other hand, we…

In this paper, we will explicitly calculate Gauss sums for the general linear groups and the special linear groups over $\Bbb Z_n$, where $\Bbb Z_n=\Bbb Z/n \Bbb Z$ and $n>0$ is an integer. For $r$ being a positive integer, the formulae of…

Let n >1 be an integer, and G a doubly transitive subgroup of the symmetric group on X={1,...,n}. In this paper we find all linear group representations rho of G on an euclidean vector space V which contains a set of equiangular vector…

Let ${\mathscr G}$ be a linear algebraic group over $k$, where $k$ is an algebraically closed field, a pseudo-finite field or the valuation ring of a nonarchimedean local field. Let $G= {\mathscr G}(k)$. We prove that if $\gamma, \delta\in…

An ideal $I$ of a commutative ring $R$ is said to be of linear type when its Rees algebra and symmetric algebra exhibit isomorphism. In this paper, we investigate the conjecture put forth by Jayanthan, Kumar, and Sarkar (2021) that if $G$…

In commutative invariant theory, a classical result due to Auslander says that if $R = \Bbbk[x_1, \dots, x_n]$ and $G$ is a finite subgroup of $\text{Aut}_{\text{gr}}(R) \cong \text{GL}(n,\Bbbk)$ which contains no reflections, then there is…

We will give a definition of quadratic forms on bimodules and prove the sandwich classification theorem for subgroups of the general linear group $\mathrm{GL}(P)$ normalized by the elementary unitary group $\mathrm{EU}(P)$ if $P$ is a…

The group classification problem for the class of (1+1)-dimensional linear $r$th order evolution equations is solved for arbitrary values of $r>2$. It is shown that a related maximally gauged class of homogeneous linear evolution equations…

We introduce the notions of relational groupoids and relational convolution algebras. We provide various examples arising from the group algebra of a group $G$ and a given normal subgroup $H$. We also give conditions for the existence of a…

A ring R is called an E-ring if the canonical homomorphism from R to the endomorphism ring End(R_Z) of the additive group R_Z, taking any r in R to the endomorphism left multiplication by r turns out to be an isomorphism of rings. In this…

We prove that if R is a principal ideal ring and A\in\M_n(R) is a matrix with trace zero, then A is a commutator, that is, A=XY-YX for some X,Y\in\M_n(R). This generalises the corresponding result over fields due to Albert and Muckenhoupt,…

Given a permutation group $G$, the derangement graph of $G$ is defined with vertex set $G$, where two elements $x$ and $y$ are adjacent if and only if $xy^{-1}$ is a derangement. We establish that, if $G$ is transitive with degree exceeding…

We study a question of Greenberg-Shalom concerning arithmeticity of discrete subgroups of semisimple Lie groups with dense commensurators. We answer this question positively for normal subgroups of lattices. This generalizes a result of the…

The Pieri rule gives an explicit formula for the decomposition of the tensor product of irreducible representation of the complex general linear group GL(n,C) with a symmetric power of the standard representation on C^n. It is an important…

We introduce the class of graded Lie-Rinehart algebras as a natural generalization of the one of graded Lie algebras. For $G$ an abelian group, we show that if $L$ is a tight $G$-graded Lie-Rinehart algebra over an associative and…

Let $G \leq \mathrm{GL}(V)$ be a group with a unique subnormal quasisimple subgroup $E(G)$ that acts absolutely irreducibly on $V$. A base for $G$ acting on $V$ is a set of vectors with trivial pointwise stabiliser in $G$. In this paper we…

In [9] Bogdan Nica presented an elementary proof of a result which says that the relative elementary linear group with respect to a square of an ideal of the ring is a subset of the true relative elementary linear group. The original result…