Related papers: Existence of magic rectangle sets over finite abel…

The aim of this small note is to prove an elementary yet useful properties of finitely presented groups. Let G be a finitely generated group with one end. Fix a (finite) generating set and let $B_n$ be the ball of radius $n$ around $e$. Let…

The concept of (a,b)-module comes from the study the Gauss-Manin lattices of an isolated singularity of a germ of an holomorphic function. It is a very simple ''abstract algebraic structure'', but very rich, whose prototype is the formal…

Given any finitely presented group G we find a triangular algebra such that has two presentations, one with fundamental group G and another with trivial group. Thus proving that given a collection G1,...,Gn of finitely presented groups…

A module is called absolutely indecomposable if it is directly indecomposable in every generic extension of the universe. We want to show the existence of large abelian groups that are absolutely indecomposable. This will follow from a more…

A conjecture in [Ish20] states that for a finite subgroup $G$ of $GL(2; \mathbb{C})$, a resolution $Y$ of $\mathbb{C}^2/G$ is isomorphic to a moduli space $\mathcal{M}_{\theta}$ of $G$-constellations for some generic stability parameter…

In this paper, we shall study finite generation of symbolic Rees rings of the defining ideal of the space monomial curves $(t^a, t^b, t^c)$ for pairwise coprime integers $a$, $b$, $c$ such that $(a,b,c) \neq (1,1,1)$. If such a ring is not…

For $G$ a finite group, one way to construct irreducible quandle representations over $\mathbb{C}$ of the conjugacy quandle $Conj(G)$ is by taking the product of an irreducible linear group representation of $G$ by what we call a quandle…

Let $G$ be a finite group, and let $V$ be a completely reducible faithful $G$-module. It has been known for a long time that if $G$ is abelian, then $G$ has a regular orbit on $V$. In this paper we show that $G$ has an orbit of size at…

Let $G=\Sp(2g,\mathbb{Z})$ be the symplectic group over the integers. Given $m\in \mathbb{N}$, it is natural to ask if there exists a non-trivial matrix $A\in G$ such that $A^{m}=I$, where $I$ is the identity matrix in $G$. In this paper,…

A lattice-ordered group (an $\ell$-group) $G(\oplus, \vee, \wedge)$ can be naturally viewed as a semiring $G(\vee,\oplus)$. We give a full classification of (abelian) $\ell$-groups which are finitely generated as semirings, by first showing…

Given a finite abelian group $G$ and elements $x, y \in G$, we prove that there exists $\phi \in \text{Aut}(G)$ such that $\phi(x) = y$ if and only if $G/\langle x \rangle \cong G/\langle y \rangle$. This result leads to our development of…

For a compact abelian group $G$, a corner in $G \times G$ is a triple of points $(x,y)$, $(x,y+d)$, $(x+d,y)$. The classical corners theorem of Ajtai and Szemer\'edi implies that for every $\alpha > 0$, there is some $\delta > 0$ such that…

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 prove a bijective unitary correspondence between 1) the isospectral torus of almost-periodic, absolutely continuous CMV matrices having fixed finite-gap spectrum and 2) special periodic block-CMV matrices satisfying a Magic Formula. This…

If a soluble group $G$ contains two finitely generated abelian subgroups $A,B$ such that the number of double cosets $AgB$ is finite, then $G$ is shown to be virtually polycyclic.

Let $\mathfrak{g}\neq \mathfrak{so}_8$ be a simple Lie algebra of type $A,D,E$ with $\widehat{\mathfrak{g}}$ the corresponding affine Kac-Moody algebra and $\mathfrak{n}_-\subset \widehat{\mathfrak{g}}$ a nilpotent subalgebra. Given…

Recognizing when a ring is a complete matrix ring is of significant importance in algebra. It is well-known folklore that a ring $R$ is a complete $n\times n$ matrix ring, so $R\cong M_{n}(S)$ for some ring $S$, if and only if it contains a…

A subset $C$ of the vertex set of a graph $\Gamma$ is said to be $(a,b)$-regular if $C$ induces an $a$-regular subgraph and every vertex outside $C$ is adjacent to exactly $b$ vertices in $C$. In particular, if $C$ is an $(a,b)$-regular set…

Let $ x $ be an element of a finite group $ G $ and denote the order of $ x $ by $ \mathrm{ord}(x) $. We consider a finite group $ G $ such that $ \gcd(\mathrm{ord}(x),\mathrm{ord}(y))\leqslant 2 $ for any two vanishing elements $ x $ and $…

For an Abelian group $G$, any homomorphism $\mu\colon G\otimes G\rightarrow G$ is called a \textsf{multiplication} on $G$. The set $\text{Mult}\,G$ of all multiplications on an Abelian group $G$ is an Abelian group with respect to addition.…