Related papers: On residually finite groups satisfying an Engel ty…

In this Note we study the groups $G$ satisfying condition $(\mathcal{N},n)$, that is, every subset of $G$ with $n+1$ elements contains a pair $\{x,y\}$ such that the subgroup $<x,y>$ is nilpotent.

We prove that there exists an integer-valued function f on positive integers such that if a finite group G has at most k real-valued irreducible characters, then |G/Sol(G)| is at most f(k), where Sol(G) denotes the largest solvable normal…

In the paper we study irreducible representations of some nilpotent groups of finite abelian total rank. The main result of the paper states that if a torsion-free minimax group $G$ of nilpotency class 2 admits a faithful irreducible…

Let $A$ be a residually finite dimensional algebra (not necessarily associative) over a field $k$. Suppose first that $k$ is algebraically closed. We show that if $A$ satisfies a homogeneous almost identity $Q$, then $A$ has an ideal of…

Let $\phi:G\rightarrow G$ be an endomorphism of a finitely generated residually finite group. R.~Hirshon asked if there exists~$n$ such that the restriction of $\phi$ to $\phi^n(G)$ is injective. We give an example to show that this is not…

We consider groups $G$ such that the set $[G,\varphi]=\{g^{-1}g^{\varphi}|g\in G\}$ is a subgroup for every automorphism $\varphi$ of $G$, and we prove that there exists such a group $G$ that is finite and nilpotent of class $n$ for every…

In the paper we show that any irreducible representation of a finitely generated nilpotent group $G$ over a finitely generated field $F$ of characteristic zero is induced from a primitive representation of some subgroup of $G$.

Let $\mathcal{C}$ be a class of finite groups closed for subgroups, quotients groups and extensions. Let $\Gamma$ be a finite simplicial graph and $G = G_{\Gamma}$ be the corresponding pro-$\mathcal C$ RAAG. We show that if $N$ is a…

A (left) Engel sink of an element g of a group G is a subset containing all sufficiently long commutators [...[[x,g],g],...,g], where x ranges over G. We prove that if p is a prime and G a finite group in which, for some positive integer m,…

We survey aspects of locally nilpotent linear groups. Then we obtain a new classification; namely, we classify the irreducible maximal locally nilpotent subgroups of $\mathrm{GL}(q, \mathbb F)$ for prime $q$ and any field $\mathbb F$.

Suppose that a locally finite group $G$ has a $2$-element $g$ with Chernikov centralizer. It is proved that if the involution in $\langle g\rangle$ has nilpotent centralizer, then $G$ has a soluble subgroup of finite index.

We prove that a subgroup of a finitely generated free nilpotent group F is undistorted if and only if it is a retract of a subgroup of finite index in F.

Let $q$ be a prime and $A$ a finite $q$-group of exponent $q$ acting by automorphisms on a finite $q'$-group $G$. Assume that $A$ has order at least $q^3$. We show that if $\gamma_{\infty} (C_{G}(a))$ has order at most $m$ for any $a \in…

A right Engel sink of an element $g$ of a group $G$ is a subset containing all sufficiently long commutators $[...[[g,x],x],\dots ,x]$. We prove that if $G$ is a compact group in which, for some $k$, every commutator $[...[g_1,g_2],\dots…

Let $g$ be an element of a finite group $G$ and let $R_{n}(g)$ be the subgroup generated by all the right Engel values $[g,{}_{n}x]$ over $x\in G$. In the case when $G$ is soluble we prove that if, for some $n$, the Fitting height of…

This work introduces and investigates the function $J(G) = \frac{\text{Nil}(G)}{L(G)}$, where $\text{Nil}(G)$ denotes the number of nilpotent subgroups and $L(G)$ the total number of subgroups of a finite group $G$. The function $J(G)$,…

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…

Let m,n be positive integers, v a multilinear commutator word and w=v^m. We prove that if G is an orderable group in which all w-values are n-Engel, then the verbal subgroup v(G) is locally nilpotent. We also show that in the particular…

Let A denote either the automorphism group of the free group of rank n>=4 or the mapping class group of an orientable surface of genus n>=12 with at most 1 boundary component, and let G be either the subgroup of IA-automorphisms or the…

We construct finitely generated Engel branch groups, answering a question of Fern\'andez-Alcober, Noce and Tracey on the existence of such objects. In particular, the groups constructed are not nilpotent, yielding the second known class of…