Related papers: Word and Conjugacy Problems in Groups $G_{k+1}^{k}…

Given a finite group $G$, denote by $\Gamma(G)$ the simple undirected graph whose vertices are the distinct sizes of noncentral conjugacy classes of $G$, and set two vertices of $\Gamma(G)$ to be adjacent if and only if they are not coprime…

Let V be a complex vector space with basis {x_1,x_2,...,x_n} and G be a finite subgroup of GL(V). The tensor algebra T(V) over the complex is isomorphic to the polynomials in the non-commutative variables x_1, x_2,..., x_n with complex…

We introduce separability properties corresponding to generalized versions of the conjugacy, twisted conjugacy, Brinkmann and Brinkmann's conjugacy problems and how they relate when finite and cyclic extensions of groups are taken. In…

The number of subgroups and the number of cyclic subgroups are natural combinatorial invariants of a finite group. We investigate how restrictions on these quantities, together with the number of distinct prime divisors of $|G|$, enforce…

Many applications donot have the benefit of the laws of physics to derive succinct descriptive models for observed data. In alternative, interdependencies among $N$ time series $\{ x_{nk}, k>0 \}_{n=1}^{N}$ are nowadays often captured by a…

Graded rings provide a natural algebraic framework for encoding symmetry via decompositions into homogeneous components indexed by a group, together with multiplication rules reflecting the group operation. Among graded rings, strongly…

In recent work on holomorphic maps that are symmetric under certain complex reflection groups---generated by complex reflections through a set of hyperplanes, the author announced a general conjecture related to reflection groups. The claim…

Let $G,D_{0},D_{1}$ be finite groups such that $D_{0}\trianglelefteq D_{1}$ are groups of automorphisms of $G$ that contain the inner automorphisms of $G$. Assume that $D_{1}/D_{0}$ has a normal $2$-complement and that $D_{1}$ acts…

In this paper a relation between iterated cyclings and iterated powers of elements in a Garside group is shown. This yields a characterization of elements in a Garside group having a rigid power, where 'rigid' means that the left normal…

William W. Boone and Graham Higman proved that a finitely generated group has soluble word problem if and only if it can be embedded in a simple group that can be embedded in a finitely presented group. We prove the exact analogue for…

A fundamental problem from invariant theory is to describe the endomorphism algebra of multilinear functions on a representation V invariant under the action of a group G. According to Weyl's classic, a first main (later: fundamental)…

This is a survey of the recent work in algorithmic and asymptotic properties of groups. I discuss Dehn functions of groups, complexity of the word problem, Higman embeddings, and constructions of finitely presented groups with extreme…

For the free group $F_r$ on $r>1$ generators (respectively, the free product $G_1 * G_2$ of two nontrivial finite groups $G_1$ and $G_2$), we obtain the asymptotic for the number of conjugacy classes of commutators in $F_r$ (respectively,…

Graph Neural Networks (GNNs) are powerful models that can manage complex data sources and their interconnection links. One of GNNs' main drawbacks is their lack of interpretability, which limits their application in sensitive fields. In…

Let $M(A,I)$ be a free partially commutative monoid with involution and $G(A,I)$ be its quotient group, e.g. a right-angled Artin or Coxeter group. Given a system of word equations over $M(A,I)$ with recognizable constraints with input size…

To study discrete dynamical systems of different types --- deterministic, statistical and quantum --- we develop various approaches. We introduce the concept of a system of discrete relations on an abstract simplicial complex and develop…

Groups with the falsification by fellow traveler property are known to have solvable word problem, but they are not known to be automatic or to have finite convergent rewriting systems. In this paper, we show that these groups admit a…

Learning to reason about relations and dynamics over multiple interacting objects is a challenging topic in machine learning. The challenges mainly stem from that the interacting systems are exponentially-compositional, symmetrical, and…

We describe the development of the theory of automatic groups. We begin with a historical introduction, define the concepts of automatic, biautomatic and combable groups, derive basic properties, then explain how hyperbolic groups and the…

Visibility algorithms are a family of geometric and ordering criteria by which a real-valued time series of N data is mapped into a graph of N nodes. This graph has been shown to often inherit in its topology non-trivial properties of the…