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Much progress has been made on the problem of calculating $g(n)$ for various classes of integers $n$, where $g$ is the group-counting function. We approach the inverse problem of solving the equations $g(n) = 6$ and $g(n) = 7$ in $n$. The…

Let a group $\Gamma$ act on a paracompact, locally compact, Hausdorff space $M$ by homeomorphisms and let $2^M$ denote the set of closed subsets of $M$. We endow $2^M$ with the Chabauty topology, which is compact and admits a natural…

By using the structure and some properties of extraspecial and generalized/almost extraspecial $p$-groups, we explicitly determine the number of elements of specific orders in such groups. As a consequence, one may find the number of cyclic…

For a functor from the category of free presentations of a group to the category of all groups we define the boundary limit as an image of the natural map from limit to colimit. We show that the fourth dimension quotient of a group can be…

We prove that if $\Gamma $ is a word hyperbolic group and $K$ is a finite subset of $\Gamma $, then $\Gamma $ admits a tile containing $K$.

A left orderable monster is a finitely generated left orderable group all of whose fixpoint-free actions on the line are proximal: the action is semiconjugate to a minimal action so that for every bounded interval $I$ and open interval $J$,…

We introduce "braided" versions of self-similar groups and R\"over--Nekrashevych groups, and study their finiteness properties. This generalizes work of Aroca and Cumplido, and the first author and Wu, who considered the case when the…

A finitely generated group $G$ is called strongly scale-invariant if there exists an injective homomorphism $f:G\to G$ such that $f(G)$ is a finite index subgroup of $G$ and such that $\cap_{n\geq 0} f^n(G)$ is finite. Nekrashevych and Pete…

Let $G$ be a Lie group with Lie algebra $\mathfrak g$. In the paper "Reality of unipotent elements in simple Lie groups, Bull. Sci. Math., 185, 2023, 103261" by K. Gongopadhyay and C. Maity, an infinitesimal version of the notion of…

We give elementary proofs of the following two theorems on automorphisms of a finite group G: (1) An automorphism of G is inner if and only if it extends to an automorphism of every finite group containing G. (2) There exists a finite…

Let $s$ be an $n$-dimensional symplectic form over a field $\mathbb{F}$ of characteristic other than $2$, with $n>2$. In a previous article, we have proved that if $\mathbb{F}$ is infinite then every element of the symplectic group…

We prove that if $G$ is a totally bounded abelian group \st\ its dual group $\widehat{G}_p$ equipped with the finite-open topology is a Baire group, then every compact subset of $G$ must be finite. This solves an open question by Chasco,…

An exhaustive version of the thesis to a talk presented at the Groups \& Algebras in Bicocca Conference (GABY), University of Milano-Bicocca (Milan, Italy), June 17 to June 21, 2024.

We give a practical computer algebra implementation of the Covering Lemma for finite transformation semigroups. The lemma states that given a surjective relational morphism $(X,S)\twoheadrightarrow(Y,T)$, we can establish emulation by a…

In this paper, we shall show that the only primitive Schur ring over a semi dihedral group is the trivial one and every semi dihedral subgroup is Burnside group, that is a primitive group containing a regular subgroup isomorphic to the semi…

In this paper we exhibit for every non amenable group that is initially sub-amenable (sometimes also referred to as LEA), two sofic approximations that are not conjugate by any automorphism of the universal sofic group. This addresses a…

The aim of this paper is to give a classification of real and strongly real unipotent elements in a classical simple Lie group. To do this, we will introduce an infinitesimal version of the notion of classical reality in a Lie group. This…

We show that there exists a quasi-isometric embedding of the product of $n$ copies of $\mathbb{H}_{\mathbb{R}}^2$ into any symmetric space of non-compact type of rank $n$, and there exists a bi-Lipschitz embedding of the product of $n$…

Consider a nonsolvable finite group G, where R(G) represents the solvable radical of G. For any element x in G, the solvabilizer of x in G, denoted by Sol_G(x), is defined as the set of all elements y in G such that the subgroup generated…

For each prime $p$ and each positive integer $d$, we construct the first examples of second countable, topologically simple, $p$-adic Lie groups of dimension $d$ whose Lie algebras are abelian. This answers several questions of Gl\"ockner…