Related papers: Infinite systems of equations in abelian and nilpo…

In this paper, we present an explicit formula for the Baer invariant of a finitely generated abelian group with respect to the variety of polynilpotent groups of class row $(c_1,...,c_t)$, ${\cal N}_{c_1,...,c_t}$. In particular, one can…

In this paper we prove unique continuation principles for some systems of elliptic partial differential equations satisfying a suitable superlinearity condition. As an application, we obtain nonexistence of nontrivial (not necessarily…

There are various results in the literature which are part of the general philosophy that a finite group for which a certain parameter (for example, the number of conjugacy classes or the maximum number of elements inverted, squared or…

We establish several finiteness properties of groups defined by algebraic difference equations. One of our main results is that a subgroup of the general linear group defined by possibly infinitely many algebraic difference equations in the…

We will show that every element of a finitely generated abelian group is automorphically equivalent what we will define to be a {\em representative element} in a {\em repeat-free subgroup}, and for finite abelian groups we can count the…

We discuss various methods and their effectiveness for solving linear equations over finitely generated abelian groups. More precisely, if $\varphi\colon G\to H$ is a homomorphism of finitely generated abelian groups and $b\in H$, we…

Let $G$ be a finite group. In this short note, we give a criterion of nilpotency of $G$ based on the existence of elements of certain order in each section of $G$.

It is shown that every separable abelian topological group is isomorphic with a topological subgroup of a monothetic group (that is, a topological group with a single topological generator). In particular, every separable metrizable abelian…

We show that every finitely generated nilpotent group of class 2 occurs as the quotient of a finitely presented abelian-by-nilpotent group by its largest nilpotent normal subgroup.

In this article, we study word equations in free semigroups and the conjecture that the existence of infinitely many solutions entails the existence of solutions with arbitrarily large exponent of periodicity. We examine this question in…

In this article we aim to develop from first principles a theory of sum sets and partial sum sets, which are defined analogously to difference sets and partial difference sets. We obtain non-existence results and characterisations. In…

We consider small nonlinear perturbations of linear systems on a time scale with the phase space being finite or infinite-dimensional. For $\Delta$-differential operators, corresponding to linear dynamic systems we consider their…

A set of quasi-uniform random variables $X_1,...,X_n$ may be generated from a finite group $G$ and $n$ of its subgroups, with the corresponding entropic vector depending on the subgroup structure of $G$. It is known that the set of entropic…

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 $…

The class of all subdirectly irreducible groups belonging to a variety generated by a finite nilpotent group can be axiomatised by a finite set of elementary sentences.

We provide polynomial lower bounds for residual finiteness of residually finite, finitely generated solvable groups that admit infinite order elements in the Fitting subgroup of strict distortion at least exponential. For this class of…

Symmetry groups allow to transform solutions of differential equations continuously into other solutions. This property can be used for the observability analysis of infinite-dimensional systems with input and output. In this contribution,…

We introduce a formalism of infinite, linearly ordered products in general groups. Using this, we define infinite compositions in certain groups of formal power series such as transseries. We show that such groups can sometimes be…

Let R be a finite unitary ring whose group of units is not solvable but all groups of units of all its proper subrings are solvable. In this paper we classify these rings and show that all finite rings of order $p^n$ for $n < 5$ and some of…

An important classical result in ZFC asserts that every infinite cardinal number is idempotent. Using this fact, we obtain several algebraic results in this article. The first result asserts that an infinite Abelian group has a proper…