Related papers: Distributive Gruppen endlicher Ordnung [Finite Dis…

We give an example of a finitely presented simple group containing a finitely generated subgroup which is not finitely presented.

This paper investigates the finite generation of cluster automorphism groups. By applying the pseudo $\mathbb{N}$-grading introduced in our previous work, we establish a sufficient condition for a cluster automorphism group to be finitely…

A homogeneous family of subsets over a given set is one with a very ``rich'' automorphism group. We prove the existence of a bi-universal element in the class of homogeneous families over a given infinite set and give an explicit…

We investigate two notions about descriptions of groups using first-order language: quasi-finite axiomatizability, concerning infinite groups, and polylogarithmic compressibility, concerning classes of finite groups.

We introduce the notion of \emph{topo-symmetric extensions} of topological groups, a new generalization of classical group extensions that incorporates both topological and symmetry constraints. We define morphisms between such extensions,…

This is a survey, intended both for group theorists and model theorists, concerning the structure of pseudofinite groups, that is, infinite models of the first order theory of finite groups. The focus is on concepts from stability theory…

Many groups possess highly symmetric generating sets that are naturally endowed with an underlying combinatorial structure. Such generating sets can prove to be extremely useful both theoretically in providing new existence proofs for…

For the first time we represent every finite group in the form of a graph in this book. The authors choose to call these graphs as identity graph, since the main role in obtaining the graph is played by the identity element of the group.…

This is an introduction to double algebras which is the structure modelled by the properties of the convolution product in Hopf algebras, weak Hopf algebras and in Hopf algebroids. We show that Hopf algebroids with a Frobenius integral can…

We describe the groups that have the same holomorph as a finite perfect group. Our results are complete for centerless groups. When the center is non-trivial, some questions remain open. The peculiarities of the general case are illustrated…

The class of finitely presented algebras over a field $K$ with a set of generators $a_{1},..., a_{n}$ and defined by homogeneous relations of the form $a_{1}a_{2}... a_{n} =a_{\sigma (1)} a_{\sigma (2)} ... a_{\sigma (n)}$, where $\sigma$…

We define a class of finite groups based on the properties of the closed twins of their power graphs and study the structure of those groups. As a byproduct, we obtain results about finite groups admitting a partition by cyclic subgroups.

In this paper, we survey some of the recent advances on embeddings into finitely generated (left-orderable) simple group such that the overgroup preserves algorithmic, geometric, or algebraic information about the embedded group. We discuss…

Let $G$ be a finite group. A number of graphs with the vertex set $G$ have been studied, including the power graph, enhanced power graph, and commuting graph. These graphs form a hierarchy under the inclusion of edge sets, and it is useful…

We outline the theory of sets with distributive operations: multishelves and multispindles, with examples provided by semi-lattices, lattices and skew lattices. For every such a structure we define multi-term distributive homology and show…

Let $\mathcal G$ denote the space of finitely generated marked groups. For any finitely generated group $G$, we construct a continuous, injective map $f$ from the space of subgroups $Sub(G)$ to $\mathcal G$ that sends conjugate subgroups to…

We introduce the concept of hyperreflection groups, which are a generalization of Coxeter groups. We prove the Deletion and Exchange Conditions for hyperreflection groups, and we discuss special subgroups and fundamental sectors of…

We consider abelain subgroups of small index in finite groups. More generally, we consider subgroups such that the product of their index by the index of their centralizer is small.

We establish several finiteness properties of groups defined by algebraic difference equations. One of our main results is that a subgroup of the general linear group defined by possibly infinitely many algebraic difference equations in the…

In this report we summarize this work, all finite simple groups $G$ can determined uniformly using their orders $|G|$ and the set $\pi_e(G)$ of their element orders.