Related papers: Probabilistically nilpotent groups

The random triangular group \Gamma(n,t) is a group given by a presentation P=<S|R>, where S is a set of n generators and R is a random set of t cyclically reduced words of length three. The asymptotic behavior of \Gamma(n,t) is in some…

Full residual finiteness growth of a finitely generated group $G$ measures how efficiently word metric $n$-balls of $G$ inject into finite quotients of $G$. We initiate a study of this growth over the class of nilpotent groups. When the…

Let G be a finite group of nilpotency class 2 and w a group word. In this short paper we show that the probability that a random n-tuple of elements from G satisfies w is at least one over the order of G. This answers a special case of a…

Let $L/K$ be a finite Galois extension of fields with group $\Gamma$. When $\Gamma$ is nilpotent, we show that the problem of enumerating all nilpotent Hopf-Galois structures on $L/K$ can be reduced to the corresponding problem for the…

Let $G$ be a group. A subset $D$ of $G$ is a determining set of $G$, if every automorphism of $G$ is uniquely determined by its action on $D$. The determining number of $G$, denoted by $\alpha(G)$, is the cardinality of a smallest…

The nilpotent graph of a group $G$ is the simple and undirected graph whose vertices are the elements of $G$ and two distinct vertices are adjacent if they generate a nilpotent subgroup of $G$. Here we discuss some topological properties of…

A group is said to be strongly amenable if each of its proximal topological actions has a fixed point. We show that a finitely generated group is strongly amenable if and only if it is virtually nilpotent. More generally, a countable…

We define a notion of roundness for finite groups. Roughly speaking, a group is round if one can order its elements in a cycle in such a way that some natural summation operators map this cycle into new cycles containing all the elements of…

Let $G$ be a group that is relatively hyperbolic with respect to a collection of subgroups $\{H_{\lambda}\}_{\lambda\in \Lambda}$. Suppose that $G$ is given by a finite relative presentation $\mathcal{P}$ with respect to this collection. We…

We present a structural description of finite nilpotent groups of class at most $2$ using a specified number of subdirect and central products of $2$-generated such groups. As a corollary, we show that all of these groups are isomorphic to…

Given a group word $w$ in $k$ variables, a finite group $G$ and $g\in G$, we consider the number $N_{w,G}(g)$ of $k$-tuples $g_1,\dots ,g_k$ of elements of $G$ such that $w(g_1,\dots ,g_k)=g$. In this work we study the functions $N_{w,G}$…

A finite semigroup is finitely related (has finite degree) if its term functions are determined by a finite set of finitary relations. For example, it is known that all nilpotent semigroups are finitely related. A nilpotent monoid is a…

It is proved that for any prime $p$ a finitely generated nilpotent group is conjugacy separable in the class of finite $p$-groups if and only if the torsion subgroup of it is a finite $p$-group and the quotient group by the torsion subgroup…

An algebra is finitely related (or has finite degree) if its term functions are determined by some finite set of finitary relations. Nilpotent monoids built from words, via Rees quotients of free monoids, have been used to exhibit many…

A group-word $w$ is concise in a class of groups $\mathcal X$ if and only if the verbal subgroup $w(G)$ is finite whenever $w$ takes only finitely many values in a group $G\in \mathcal X$. It is a long-standing open problem whether every…

We extend and reformulate a result of Solomon on the divisibility of the title. We show, for example, that if $\Gamma$ is a finitely generated group, then $|G|$ divides $#\Hom(\Gamma,G)$ for every finite group $G$ if and only if $\Gamma$…

A group is small if it has countably many complete $n$-types over the empty set for each natural number n. More generally, a group $G$ is weakly small if it has countably many complete 1-types over every finite subset of G. We show here…

We study the groups $G$ with the curious property that there exists an element $k\in G$ and a function $f\colon G\to G$ such that $f(xk)=xf(x)$ holds for all $x\in G$. This property arose from the study of near-rings and input-output…

In this paper, we study nilpotent $\mathbb{Q}$$[x]$-powered groups that satisfy the following property: For some set of primes $\omega$ in $\mathbb{Q}$$[x]$, every $\omega '$-isolated $\mathbb{Q}$$[x]$-subgroup in some family of its…

For a finite group $G$ and a non-negative integer $d$, denote by $\Omega_d(G)$ the number of functions $G^d\rightarrow G$ that are induced by substitution into a word with variables among $X_1,\ldots,X_d$. In this note, we show that…