Related papers: Artinian groups of large cardinality

We introduce a unified theory of Cartan subgroups and maximal toroids - defined as connected multiplicative type subgroups that are maximal amongst all such subgroups - which holds for all affine algebraic groups over a field, regardless of…

A proper subgroup $H$ in a finite group $G$ is said to be large if $|H|^3\geq |G|$. In this paper, we determined all large maximal subgroups of almost simple classical groups. Combined with the work of Alavi and Burness (J. Algebra 421…

We show that the membership problem in a finitely generated submonoid of a graph group (also called a right-angled Artin group or a free partially commutative group) is decidable if and only if the independence graph (commutation graph) is…

Cluster algebras are a recent topic of study and have been shown to be a useful tool to characterize structures in several knowledge fields. An important problem is to establish whether or not a given cluster algebra is of finite type.…

Let A denote the algebraic closure of the rationals Q in the complex numbers C. Suppose G is a torsion-free group which contains a congruence subgroup as a normal subgroup of finite index and denote by U(G) the C-algebra of closed densely…

A concrete family of automorphisms alpha_n of the free group F_n is exhibited, for any n > 2, and the following properties are proved: alpha_n is irreducible with irreducible powers, has trivial fixed subgroup, and has 2n-1 attractive as…

We conjecture that every infinite group $G$ can be partitioned into countably many cells $G=\bigcup_{n\in\omega}A_n$ such that $cov(A_nA_n^{-1})=|G|$ for each $n\in\omega$. Here $cov(A)=\min\{|X|:X\subseteq G, G=XA\}$. We confirm this…

Among all affine, flat, finitely presented group schemes, we focus on those that are pure, this includes all groups which are extensions of a finite locally free group by a group with connected fibres. We prove that over an arbitrary base…

Given a Lie algebra $L$ graded by a group $G$, if $L$ is does not contain orthogonal graded ideals and $G$ is generated by the support of $L$, then $G$ is an abelian group.

A set of elements of a finite abelian group is called sum-free if it contains no Schur triple, i.e., no triple of elements $x,y,z$ with $x+y=z$. The study of how large the largest sum-free subset of a given abelian group is had started more…

Following a previous work with Boudi, we continue to investigate Bernstein algebras satisfying chain conditions. First, it is shown that a Bernstein algebra $(A, \omega)$ with ascending or descending chain condition on subalgebras is…

We construct examples of localizations in the category of groups which take the Mathieu group $M_{11}$ to groups of arbitrarily large cardinality which are ``abelian up to finitely many generators''. The paper is part of a broader study on…

We prove that non-abelian free groups of finite rank at least 3 or of countable rank are not $\forall$-homogeneous. We answer three open questions from Kharlampovich, Myasnikov, and Sklinos regarding whether free groups, finitely generated…

We show that an infinite residually finite boundedly generated group has an infinite chain of finite index subgroups with ranks uniformly bounded, and give (sublinear) upper bounds on the ranks of arbitrary finite index subgroups of…

We show that every Abelian group satisfying a mild cardinal inequality admits a pseudocompact group topology from which all countable subgroups inherit the maximal totally bounded topology (we say that such a topology satisfies property…

We describe a family of rational affine surfaces S with huge groups of automorphisms in the following sense: the normal subgroup of Aut(S) generated by all its algebraic subgroups is not generated by any countable family of such subgroups,…

We answer a long-standing open question by proving in ordinary set theory, ZFC, that the Kaplansky test problems have negative answers for aleph_1-separable abelian groups of cardinality aleph_1. In fact, there is an aleph_1-separable…

We consider a class K of structures e.g. trees with omega +1 levels, metric spaces and mainly, classes of Abelian groups like the one mentioned in the title and the class of reduced separable (Abelian) p-groups. We say M in K is universal…

We show that free Burnside groups of sufficiently large odd exponent are non--amenable in a certain strong sense, more precisely, their left regular representations are isolated from the trivial representation uniformly on finite generating…

An algebra is called a GI-algebra if its group of units satisfies a group identity. We provide positive support for the following two open problems. 1. Does every algebraic GI-algebra satisfy a polynomial identity? 2. Is every algebraically…