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We show that there is an order-preserving embedding of the additive group of rational numbers $\mathbb{Q}$ into a 2-generator group $G$. The group $G$ can be chosen to be a solvable group $G$ of length 3, which is a minimal result in the…

This communication records some observations made in the course of studying one-relator groups from the point of view of residual solvability. As a contribution to clas- sification efforts we single out some relator types that render the…

We introduce and study some families of groups whose irreducible characters take values on quadratic extensions of the rationals. We focus mostly on a generalization of inverse semi-rational groups, which we call uniformly semi-rational…

Let G be a totally disconnected, locally compact group. A closed subgroup of G is locally normal if its normaliser is open in G. We begin an investigation of the structure of the family of closed locally normal subgroups of G. Modulo…

Let $G$ be a connected reductive group split over R. We show that every unipotent element in the totally nonnegative monoid of G is regular in some Levi subgroups, confirming a conjecture of Lusztig.

We prove a conjecture due to Baumgaertel and Lledo according to which for every compact group G one has Z(G)^ \cong C(G), where the `chain group' C(G) is the free abelian group (written multiplicatively) generated by the set G^ of…

Let $V$ be a finite-dimensional vector space over a finite field, and suppose $G \leq \Gamma \mathrm{L}(V)$ is a group with a unique subnormal quasisimple subgroup $E(G)$ that is absolutely irreducible on $V$. A base for $G$ is a set of…

Let $G$ be a unimodular locally compact group. We define a property of irreducible unitary $G$-representations $V$ which we call c-temperedness, and which for the trivial $V$ boils down to F{\o}lner's condition (equivalent to the trivial…

If $K$ is a closed subgroup of a compact group $G$, the probability that randomly chosen pair of elements from $K$ and $G$ commute is denoted by $Pr(K,G)$. Say that a subgroup $K\leq G$ is $\epsilon$-central in $G$ if $Pr(\langle g…

A non-trivial element of a group is a generalized torsion element if some products of its conjugates is the identity. The minimum number of such conjugates is called a generalized torsion order. We provide several restrictions for…

We prove that the $k$th term of the lower central series of a finite group $G$ is nilpotent if and only if $|ab|=|a||b|$ for any $\gamma_k$-commutators $a,b\in G$ of coprime orders.

Let V be a simple vertex operator algebra and let G be a finite automorphism group of V. In [DY], it was shown that any irreducible V-module is a completely reducible V^G-module where V^G is the G-invariant sub-vertex operator algebra of V.…

The aim of this brief note is to offer another proof of a theorem of Hrushovski that approximate subgroups of GL_n(C) are almost nilpotent. This approach generalizes to uniformly non amenable groups.

We study a characteristic subgroup of finitely generated groups, consisting of elements with uniform upper bound for word-lengths. For a group $G$, we denote this subgroup by $G_{bound}$. We give sufficient criteria for triviality and…

In this article, we study the normal generation of the mapping class group. We first show that a mapping class is a normal generator if its restriction on the invariant subsurface normally generates the (pure) mapping class group of the…

Let $G$ be a connected closed subgroup of $\mathrm{GL}_n(\mathbb{C})$ which is simple as a Lie group and which acts irreducibly on $\mathbb{C}^n$. Regarding both $G$ and its Lie algebra $\mathfrak{g}$ as subsets of $M_n(\mathbb{C})$, we…

Let $G$ be a finite group and $N_{\Omega}(G)$ be the intersection of the normalizers of all subgroups belonging to the set $\Omega(G),$ where $\Omega(G)$ is a set of all subgroups of $G$ which have some theoretical group property. In this…

Let $G$ be a connected reductive subgroup of a complex connected reductive group $\hat{G}$. Fix maximal tori and Borel subgroups of $G$ and $\hat{G}$. Consider the cone $LR^\circ(\hat{G},G)$ generated by the pairs $(\nu,\hat{\nu})$ of…

Let $w=w(x_1,...,x_n)$ be a word, i.e. an element of the free group $F = \langle x_1,...,x_n \rangle$. The verbal subgroup $w(G)$ of a group $G$ is the subgroup generated by the set $\{ w(x_1,...,x_n) : x_1,...,x_n \in G \}$ of all…

Given a finite group $G$, we denote by $\nu(G)$ the probability that two randomly chosen elements of $G$ generate a nilpotent subgroup. We prove that if $\nu(G)>1/12,$ then $G$ is solvable.