Related papers: Compact groups in which all elements have countabl…

We show that every definable group G in an o-minimal structure is definably finitely generated. That is, G contains a finite subset that is not included in any proper definable subgroup. This provides another proof, and a generalization to…

In this article it is proved, that every locally compact second countable group has a left invariant metric d, which generates the topology on G, and which is proper, ie. every closed d-bounded set in G is compact. Moreover, we obtain the…

Let $G$ be a finite group and $H$ be a subgroup of $G$. In this paper, we prove that if $G$ is a finite nilpotent group and $H$ a subgroup of $G$, then $H$ is normal in $G$ if and only if all normalized right transversals of $H$ in $G$ are…

Let $G$ be an infinite locally compact group and $\aleph$ a cardinal satisfying $\aleph_0\le\aleph\le w(G)$ for the weight $w(G)$ of $G$. It is shown that there is a closed subgroup $N$ of $G$ with $w(N)=\aleph$. Sample consequences are:…

Let $F$ be a finitely generated free group and let $H\le F$ be a finitely generated subgroup. Given an element $g\in F$, we study the ideal $\mathfrak{I}_g$ of equations for $g$ with coefficients in $H$, i.e. the elements $w(x)\in H*\langle…

If $S$ is a discrete semigroup, then $\beta S$ has a natural, left-topological semigroup structure extending $S$. Under some very mild conditions, $U(S)$, the set of uniform ultrafilters on $S$, is a two-sided ideal of $\beta S$, and…

If $H$ is a subgroup of a compact group $G$, the probability that a random element of $H$ commutes with a random element of $G$ is denoted by $Pr(H,G)$. Let $\langle g\rangle$ stand for the monothetic subgroup generated by an element $g\in…

Suppose that a locally finite group $G$ has a $2$-element $g$ with Chernikov centralizer. It is proved that if the involution in $\langle g\rangle$ has nilpotent centralizer, then $G$ has a soluble subgroup of finite index.

We prove that no uncountable Polish group can admit a system of generators whose associated length function satisfies the following conditions: (i) if $0 < k < \omega$, then $lg(x) \leq lg(x^k)$; (ii) if $lg(y) < k < \omega$ and $x^k = y$,…

A finite group $G$ is said to be rational if every character of $G$ is rational-valued. The Gruenberg-Kegel graph of a finite group $G$ is the undirected graph whose vertices are the primes dividing the order of $G$ and the edges join…

Over each nontrivial finite group $G$, there exists a finite system of equations having no solutions in larger finite groups but having a solution in a periodic group containing $G$. We prove several similar facts about amenable, orderable,…

An $integral$ of a group $G$ is a group $H$ whose derived group (commutator subgroup) is isomorphic to $G$. This paper discusses integrals of groups, and in particular questions about which groups have integrals and how big or small those…

Let $G$ be a locally profinite group and let $k$ be a field of positive characteristic $p$. Let $Z(G)$ denote the center of $G$ and let $\mathfrak{Z}(G)$ denote the Bernstein center of $G$, that is, the $k$-algebra of natural endomorphisms…

We complete the study of finite and profinite groups admitting an action by an elementary abelian group under which the centralizers of automorphisms consist of Engel elements. In particular, we prove the following theorems. Let $q$ be a…

A topological group $G$ is extremely amenable if every continuous action of $G$ on a compact space has a fixed point. Using the concentration of measure techniques developed by Gromov and Milman, we prove that the group of automorphisms of…

A finite order element $g$ of a group $G$ is called rational if $g$ is conjugate to $g^i$ for every integer $i$ coprime to the order $g$. We determine all triples $(G,g,\phi)$, where $G$ is a simple algebraic group of type $A_n,B_n$ or…

Given any pair of countable groups $G$ and $H$ with $G$ infinite, we construct a minimal, free, Cantor $G$-flow $X$ so that $H$ embeds into the group of automorphisms of $X$.

For any odd prime $p$, we give an example of a locally finite $p$-group $G$ containing a left 3-Engel element $x$ where $\langle x \rangle^G$ is not nilpotent.

We show that every product of f.g.\ submonoids of a group $G$ is a section of a f.g.\ submonoid of $G{\times}H_5(\mathbb{Z})$, where $H_5(\mathbb{Z})$ is a Heisenberg group. This gives us a converse of a reduction of Bodart, and a new…

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…