Related papers: On triple factorisations of finite groups

Rational transformations of polynomials are extensively studied in the context of finite fields, especially for the construction of irreducible polynomials. In this paper, we consider the factorization of rational transformations with…

We introduce a new approach to an enumerative problem closely linked with the geometry of branched coverings; that is, we study the number of ways a permutation can be decomposed into a product of a given number of 2-cycles, 3-cycles, etc.…

A notion of degeneration of elements in groups is introduced. It is used to parametrize the orbits in a finite abelian group under its full automorphism group by a finite distributive lattice. A pictorial description of this lattice leads…

Recently, a number of interesting relations have been discovered between generalised Pauli/Dirac groups and certain finite geometries. Here, we succeeded in finding a general unifying framework for all these relations. We introduce…

Let $G$ be a finite group and $\alpha(G)=\frac{|C(G)|}{|G|}$\,, where $C(G)$ denotes the set of cyclic subgroups of $G$. In this short note, we prove that $\alpha(G)\leq\alpha(Z(G))$ and we describe the groups $G$ for which the equality…

We introduce notions of absolutely non-free and perfectly non-free group actions and use them to study the associated unitary representations. We show that every weakly branch group acts absolutely non-freely on the boundary of the…

We investigate near-factorizations of nonabelian groups, concentrating on dihedral groups. We show that some known constructions of near-factorizations in dihedral groups yield equivalent near-factorizations. In fact, there are very few…

For every natural number k we introduce the notion of k-th order convolution of functions on abelian groups. We study the group of convolution preserving automorphisms of function algebras in the limit. It turns out that such groups have…

Let $A$ be a rational function of degree $n\geq 2$. Let us denote by $ G(A)$ the group of M\"obius transformations $\sigma$ such that $ A\circ \sigma=\nu_{\sigma} \circ A$ for some M\"obius transformations $\nu_{\sigma}$, and by $\Sigma(A)$…

Let $G$ be a finite group having a factorisation $G=AB$ into subgroups $A$ and $B$ with $B$ cyclic and $A\cap B=1,$ and let $b$ be a generator of $B$. The associated skew-morphism is the bijective mapping $f:A \to A$ well defined by the…

Let $G$ be a semisimple algebraic group. We develop a machinery for manipulation and manufacture of well-rounded families $\left\{ \mathcal{B}_{T}\right\} _{T>0}\subset G$ as they were defined in a work by A. Gorodnik and A. Nevo. The…

High order calculation at semi-hard scale is very important, but a satisfactory calculation framework is still missing. We propose a systematic method to regularize rapidity divergences in the CGC factorization, which makes higher order…

We study certain actions of finitely generated abelian groups on higher dimensional noncommutative tori. Given a dimension $d$ and a finitely generated abelian group $G$, we apply a certain function to detect whether there is a simple…

The theorem of factorisation forests shows the existence of nested factorisations -- a la Ramsey -- for finite words. This theorem has important applications in semigroup theory, and beyond. The purpose of this paper is to illustrate the…

In this paper, we characterize finite group $G$ with unique proper non-abelian element centralizer. This improves \cite[Theorem 1.1]{nab}. Among other results, we have proved that if $C(a)$ is the proper non-abelian element centralizer of…

We extend \cite{G} to the nonsemisimple case. We define and study exact factorizations $\B=\A\bullet \C$ of a finite tensor category $\B$ into a product of two tensor subcategories $\A,\C\subset \B$, and relate exact factorizations of…

For a group $G$ and a finite set $A$, denote by $\text{End}(A^G)$ the monoid of all continuous shift commuting self-maps of $A^G$ and by $\text{Aut}(A^G)$ its group of units. We study the minimal cardinality of a generating set, known as…

We determine the finite groups $G$ in which every subset $A \subseteq G$ of cardinality dividing the order of $G$ is a \emph{factor}, i.e. has a complement $B \subseteq G$ of cardinality $|G|/|A|$ such that $G = A \cdot B$ or $G = B \cdot…

Given an action of an affine algebraic group with only trivial characters on a factorial variety, we ask for categorical quotients. We characterize existence in the category of algebraic varieties. Moreover, allowing constructible sets as…

We give a classification of all quasitriangular structures and ribbon elements of $\mathcal{D}(G)$ explicitly in terms of group homomorphisms and central subgroups. This can equivalently be interpreted as an explicit description of all…