Related papers: Totally deranged elements of almost simple groups …

A group G is almost cyclic if there is an element x in G, such that for all g in G, there is an element y in G and an integer n with ygy^{-1} = x^n (that is, every element is conjugate to some power of x). W. Ziller asked whether there are…

We say that an element $g$ of a group $G$ is almost right Engel if there is a finite set ${\mathscr R}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$, that is, for…

It is shown that for any action of a finitely presented group $G$ on an $\R$-tree, there is a decomposition of $G$ as the fundamental group of a graph of groups related to this action. If the action of $G$ on $T$ is non-trivial, i.e. there…

The semidirect product $\mathbb{G}=\mathbb{L}\rtimes \mathbb{K}$ attached to a compact-group action on a connected, simply-connected solvable Lie group has a dense set of compact elements precisely when the $s\in \mathbb{K}$ operating on…

We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If k is the minimal degree of a representation of the finite group G, then for every subset B of G with $|B| > |G| / k^{1/3}$ we have B^3 =…

A semigroup is completely simple if it has no proper ideals and contains a primitive idempotent. We say that a completely simple semigroup $S$ is a homogeneous completely simple semigroup if any isomorphism between finitely generated…

We say that a finite group $G$ satisfies the independence property if, for every pair of distinct elements $x$ and $y$ of $G$, either $\{x,y\}$ is contained in a minimal generating set for $G$ or one of $x$ and $y$ is a power of the other.…

Let $G$ be a simple algebraic group over an algebraically closed field $K$ of characteristic $p > 0$. We consider connected reductive subgroups $X$ of $G$ that contain a given distinguished unipotent element $u$ of $G$. A result of…

We consider the class $\mathcal{A}_0$ of Abelian block-rigid $CRQ$-groups of ring type. A subgroup $A$ of an Abelian group $G$ is called an \textsf{absolute ideal} of the group $G$ if $A$ is an ideal in any ring on $G$. We describe…

Let $G$ be a group. Write $G^{*}=G\setminus \{1\}$. An element $x$ of $G^{*}$ will be called deficient if $ \langle x\rangle < C_G(x)$ and it will be called non-deficient if $\langle x\rangle = C_G(x).$ If $x\in G$ is deficient…

Let $n>0$ be an integer and $\mathcal{X}$ be a class of groups. We say that a group $G$ satisfies the condition $(\mathcal{X},n)$ whenever in every subset with $n+1$ elements of $G$ there exist distinct elements $x,y$ such that $<x,y>$ is…

Let $G$ be a simple algebraic group of exceptional type over an algebraically closed field of characteristic $p \geqslant 0$ which is not algebraic over a finite field. Let $\mathcal{C}_1, \ldots, \mathcal{C}_t$ be non-central conjugacy…

If $\G$ is a finitely generated group with generators $\{g_1,...,g_j\}$ then an infinite order element $f \in \G$ is a {\em distortion element} of $\G$ provided $\displaystyle{\liminf_{n \to \infty} |f^n|/n = 0,}$ where $|f^n|$ is the word…

Let $X$ be a smooth manifold belonging to one of these three collections: acyclic manifolds (compact or not, possibly with boundary), compact connected manifolds (possibly with boundary) with nonzero Euler characteristic, integral homology…

The article continues the series of papers on the absolute of finitely generated groups. The absolute of a group with a fixed system of generators is defined as the set of ergodic Markov measures for which the system of cotransition…

For every group $G$, the set $\mathcal{P}(G)$ of its subsets forms a semiring under set-theoretical union $\cup$ and element-wise multiplication $\cdot$ and forms an involution semigroup under $\cdot$ and element-wise inversion ${}^{-1}$.…

A subset of a group invariably generates the group if it generates even when we replace the elements by any of their conjugates. In a 2016 paper, Pemantle, Peres and Rivin show that the probability that four randomly selected elements…

A measure preserving action of a countably infinite group \Gamma is called totally ergodic if every infinite subgroup of \Gamma acts ergodically. For example, all mixing and mildly mixing actions are totally ergodic. This note shows that if…

A finite group $G$ is said to have the nilpotent decomposition property (ND) if for every nilpotent element $\alpha$ of the integral group ring $\mathbb{Z}[G]$ one has that $\alpha e$ also belong to $\mathbb{Z}[G]$, for every primitive…

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…