Related papers: Finiteness properties of affine difference algebra…

We develop the theory of difference algebraic groups in the case where we have finitely many pairwise commuting difference operators. We show that the defining ideal of a difference algebraic group is finitely generated as a difference…

Classes of algebraic structures that are defined by equational laws are called varieties or equational classes. A variety is finitely generated if it is defined by the laws that hold in some fixed finite algebra. We show that every…

Broadly speaking, a finiteness property of groups is any generalisation of the property of having finite order. A large part of infinite group theory is concerned with finiteness properties and the relationships between them. Profinite…

We look at spaces of infinite-by-infinite matrices, and consider closed subsets that are stable under simultaneous row and column operations. We prove that up to symmetry, any of these closed subsets is defined by finitely many equations.

Recently there has been a lot of research and progress in profinite groups. We survey some of the new results and discuss open problems. A central theme is decompositions of finite groups into bounded products of subsets of various kinds…

Finite difference schemes are here solved by means of a linear matrix equation. The theoretical study of the related algebraic system is exposed, and enables us to minimize the error due to a finite difference approximation.

This survey describes some recent work, by the authors and others, on the existence of algebraic fibrations of group extensions, as well as the finiteness properties of their algebraic fibers, in the realm of both abstract and pro-$p$…

Affine difference algebraic groups are a generalization of affine algebraic groups obtained by replacing algebraic equations with algebraic difference equations. We show that the isomorphism theorems from abstract group theory have…

Arithmetical properties of a finite group are properties of the group which are defined by its arithmetical parameters such as the order of the group, the element orders and so on. In this paper, we discuss a number of results on…

Several results on presenting an affine algebraic group variety as a product of algebraic varieties are obtained.

For the coordinate algebras of connected affine algebraic groups, we explore the problem of finding a presentation by generators and relations canonically determined by the group structure.

We describe gradings by finite abelian groups on the associative algebras of infinite matrices with finitely many nonzero entries, over an algebraically closed field of characteristic zero.

In this note we introduce and characterize a class of finite groups for which the element orders satisfy a certain inequality. This is contained in some well-known classes of finite groups.

We determine the finite groups whose real irreducible representations have different degrees.

We construct an explicit filtration of the ring of algebraic power series by finite dimensional constructible sets, measuring the complexity of these series. As an application, we give a bound on the dimension of the set of algebraic power…

In this paper we study the possibility to define irreducible representations of the symmetric groups with the help of finitely many relations. The existence of finite bases is established for the classes of representations corresponding to…

A recurring theme in finite group theory is understanding how the structure of a finite group is determined by the arithmetic properties of group invariants. There are results in the literature determining the structure of finite groups…

We explore connected affine algebraic groups $G$, which enjoy the following finiteness property $\rm (F)$: for every algebraic action of $G$, the closure of every $G$-orbit contains only finitely many $G$-orbits. We obtain two main results.…

Let $d \geq 2$ be an integer. We conjecture that there is a finitely generated perfect group whose homomorphic images include all finite $d$-generated perfect groups. We prove a special case of this conjecture for the finite perfect groups…

We construct the first example of a finitely-presented, residually-finite group that contains an infinite sequence of non-isomorphic finitely-presented subgroups such that each of the inclusion maps induces an isomorphism of profinite…