Related papers: Purely noncommuting groups

A combing is a set of normal forms for a finitely generated group. This article investigates the language-theoretic and geometric properties of combings for nilpotent and polycyclic groups. It is shown that a finitely generated class 2…

We introduce a notion of noncommutative Choquet simplex, or briefly an nc simplex, that generalizes the classical notion of a simplex. While every simplex is an nc simplex, there are many more nc simplices. They arise naturally from…

We study groups having the property that every non-abelian subgroup contains its centralizer. We describe various classes of infinite groups in this class, and address a problem of Berkovich regarding the classification of finite $p$-groups…

The question of whether there exists a finite group of order at least three in which every element except one is a commutator has remained unresolved in group theory. In this article, we address this open problem by developing an…

A finitely generated group $G$ is said to be condensed if its isomorphism class in the space of finitely generated marked groups has no isolated points. We prove that every product variety $\mathcal{UV}$, where $\mathcal{U}$ (respectively,…

For any topological group $G$ the dual object $\hat G$ is defined as the set of equivalence classes of irreducible unitary representations of $G$ equipped with the Fell topology. If $G$ is compact, $\hat G$ is discrete, and we investigate…

We construct finitely generated torsion-free solvable groups $G$ that have infinite rank, but such that all finitely generated torsion-free metabelian subquotients of $G$ are virtually abelian. In particular all finitely generated…

Let $G$ be a nonabelian group. We say that $G$ has an abelian partition, if there exists a partition of $G$ into commuting subsets $A_1, A_2, \ldots, A_n$ of $G$, such that $|A_i|\geqslant 2$ for each $i=1, 2, \ldots, n$. This paper…

In this note, it is proved the existence of an infinitely generated multiplicative group consisting of entire functions that are, except for the constant function 1, hypercyclic with respect to the convolution operator associated to a given…

We study actions of groups by homeomorphisms on $\mathbf{R}$ (or an interval) that are minimal, have solvable germs at $\pm \infty$ and contain a pair of elements of a certain type. We call such actions coherent. We establish that such an…

We prove that every acylindrically hyperbolic group that has no non-trivial finite normal subgroup satisfies a strong ping pong property, the $P_{naive}$ property: for any finite collection of elements $h_1, \dots, h_k$, there exists…

Let $H$ be a subgroup of a group $G$. The permutizer $P_G(H)$ is the subgroup generated by all cyclic subgroups of $G$ which permute with $H$. A subgroup $H$ of a group $G$ is strongly permutable in $G$ if $P_U(H)=U$ for every subgroup $U$…

Let $G$ be a finite, non-abelian group of the form $G = A N$, where $A \leq G$ is abelian, and $N \trianglelefteq G$ is cyclic. We prove that the commuting graph $\Gamma(G)$ of $G$ is either a connected graph of diameter at most four, or…

We study algebraic properties of groups of PL or smooth homeomorphisms of unit cubes in any dimension, fixed pointwise on the boundary, and more generally PL or smooth groups acting on manifolds and fixing pointwise a submanifold of…

Just infinite groups play a significant role in profinite group theory. For each $c \geq 0$, we consider more generally JNN$_c$F profinite (or, in places, discrete) groups that are Fitting-free; these are the groups $G$ such that every…

We study a commutation pattern in which two affine families commute completely across the two families while each family retains internal noncommutativity. For one-dimensional affine groups over finite commutative rings, we prove a…

A countable group is C*-simple if its reduced C*-algebra is a simple algebra. Since Powers recognised in 1975 that non-abelian free groups are C*-simple, large classes of groups which appear naturally in geometry have been identified,…

We introduce the class of permawound unipotent groups, and show that they simultaneously satisfy certain "ubiquity" and "rigidity" properties that in combination render them very useful in the study of general wound unipotent groups. As an…

We prove that every free metabelian non--cyclic group has a finitely generated isolated subgroup which is not separable in the class of nilpotent groups. As a corollary we prove that for every prime number $p$ an arbitrary free metabelian…

A transitive permutation group is semiprimitive if each of its normal subgroups is transitive or semiregular. Interest in this class of groups is motivated by two sources: problems arising in universal algebra related to collapsing monoids…