群论

Let $L$ be one of the finite simple classical groups $L_8(q)$, $U_8(q)$, $O_{10}^+(q)$, $O_{10}^-(q)$ or $O_{12}^+(q)$, with $q$ odd. We prove that every finite group having the same set of element orders as $L$ is an almost simple group…

We introduce the concept of relational depth of a finite semigroup $S$ whose $J$-classes form a chain. It captures how far down in the ideal structure one is obliged to go in order to define the semigroup by generators and defining…

Using the fact that Hopf-Galois structures on separable extensions and skew bracoids are both intrinsically connected to transitive subgroups of the holomorph of a finite group, we present algorithms to classify and enumerate these objects…

The spectrum of a finite group is the set of orders of its elements. We are concerned with finite groups having the same spectrum as a direct product of nonabelian simple groups with abelian Sylow $2$-subgroups. For every positive integer…

We give an explicit description of the set of element orders for every almost simple group with socle $E_7(q)$.

We investigate the properties of the intersection $\mathrm{Int}_{\mathfrak{F}}(G)$ of all $\mathfrak{F}$-maximal subgroups of a finite group $G$ for a hereditary formation $\mathfrak{F}$ of finite groups. We prove that…

Given a strict partial order $\Delta$ on a set $\Lambda$ and an arbitrary ring $R$ with $1\neq 0$, the corresponding McLain group $M(\Delta)$ has been studied in depth. We construct a larger family of McLain groups $G(\Delta)$, where…

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…

Let $G$ be a linear semisimple Lie group without compact factors. We show that uniform approximate lattices $\Lambda$ arising as regular model sets in $G$ determine the ambient group $G$ in a strong sense. Specifically, for every…

We prove that there is a unique graph with four edges which is the Gruenberg-Kegel graph of a solvable cut group. This contributes to the classification of the Gruenberg-Kegel graphs of solvable cut groups initiated in [A. B\"achle, A.…

The 5-element Brandt semigroup $B_2$ admits the structure of a naturally semilattice-ordered inverse semigroup, thus becoming an additively idempotent semiring with the operation of taking greatest lower bounds as the semiring addition. For…

Given a group $G$ acting faithfully on a set $S$, one gets a simple group denoted $SV_G$, called a twisted Brin--Thompson group. In this paper we drop the faithfulness assumption, and get an abstract version of a twisted Brin--Thompson…

We study equalizers and fixed points of monomorphisms of free groups at infinity. We show that the action of the equalizer of two monomorphisms on the regular points of the equalizer at infinity has finitely many orbits, showing that the…

Thompson proved that every finite group admitting a fixed-point-free automorphism of prime order is nilpotent, and Kegel showed that the same conclusion holds for finite groups admitting a splitting automorphism of prime order. Motivated by…

We provide a general construction of convex cocompact hyperbolic reflection groups with three-dimensional limit sets. More precisely, our construction takes as input an arbitrary simplicial complex L of dimension 3 on n vertices, and…

In this note we provide a negative answer to the question: ``Is it true that for every positive rational number $r$ there exists a finite abelian group $G$ such that $|\mathrm{Aut}(G)|/|G| = r$?". We show that if $r = a/b$ is a rational…

Bass-Serre theory provides a powerful framework for studying group actions on trees. While extremely effective for structural questions in group theory, it is less suited to the systematic construction of group actions with prescribed local…

We construct two families of examples of pro-p groups, with rather elementary presentations, that do not complete into 1-cyclotomic oriented pro-p groups. These provide brand new examples of pro-p groups that do not occur as maximal pro-p…

We study a conjecture by Deaconescu on the solubility of finite groups with claims that if more than half of the elements in a finite group has the same order $k$, then the group is soluble. We show that the original conjecture fails by…

We extend the class of abelian groups for which a conjecture of Asai and Yoshida on the number of crossed homomorphisms holds. We also prove a general result which connects certain problems concerning divisibility in groups to the…