Related papers: Certain locally nilpotent varieties of groups

Let $A$ be an elementary abelian group of order $p^{k}$ with $k\geq 3$ acting on a finite $p'$-group $G$. The following results are proved. If $\gamma_{k-2}(C_{G}(a))$ is nilpotent of class at most $c$ for any $a\in A^{#}$, then…

In this paper, we show that a locally graded group with a finite number m of non-(nilpotent of class at most n) subgroups is (soluble of class at most [log2(n)] + m + 3)-by-(finite of order $\le$ m!). Also we show that the derived length of…

This paper deals with the variety of commutative nonassociative algebras satisfying the identity $L_x^3+ \gamma L_{x^3} = 0$, $\gamma \in K$. Correa et al proved that if $\gamma = 0,1$ then any such finitely generated algebra is nilpotent.…

We call a group $G$ nilpotently Jordan of class at most $c$ $(c\in\mathbb{N})$ if there exists a constant $J\in\mathbb{Z}^+$ such that every finite subgroup $H\leqq G$ contains a nilpotent subgroup $K\leqq H$ of class at most $c$ and index…

Let $G$ be a locally graded group and suppose that every non-nilpotent subgroup of $G$ is permutable. We prove that $G$ is soluble. (In light of previous results of the authors, it suffices to prove that $G$ is soluble if it is periodic.

Let $D$ be a division ring with the center $F$ and $D^*$ be the multiplicative group of $D$. In this paper we study locally nilpotent maximal subgroups of $D^*$. We give some conditions that influence the existence of locally nilpotent…

In this paper we present some inequalities for the order, the exponent, and the number of generators of the c-nilpotent multiplier (the Baer invariant with respect to the variety of nilpotent groups of class at most $c \geq 1$) of a…

In this paper we present some inequalities for the order, the exponent and the number of generators of the $c$-nilpotent multiplier (the Baer invariant with respect to the variety of nilpotent groups of class at most $c \geq 1$) of a finite…

We provide the first example of virtually nilpotent group, with a specific generating set, for which the Green series (sometimes called cogrowth series) is not $D$-finite. The proof relies on an arithmetical miracle, and the study of the…

Let $G$ be a finite group, and $c$ an element of $\mathbb{Z}\cup \{\infty\}$. A subgroup $H$ of $G$ is said to be {\it $c$-nilpotent} if it is nilpotent, and has nilpotency class at most $c$. A subset $X$ of $G$ is said to be {\it…

Let $q$ be a prime. Let $G$ be a residually finite group satisfying an identity. Suppose that for every $x \in G$ there exists a $q$-power $m=m(x)$ such that the element $x^m$ is a bounded Engel element. We prove that $G$ is locally…

The authors have shown previously that every locally pro-p contraction group decomposes into the direct product of a p-adic analytic factor and a torsion factor. It has long been known that p-adic analytic contraction groups are nilpotent.…

We offer in this note a self-contained proof of the fact that a finitely generated group is not virtually nilpotent if and only if it has a quotient with the infinite conjugacy class (ICC) propoerty. This proof is a modern presentation of…

Denote by C_{n,d} the nilpotency degree of a relatively free algebra generated by d elements and satisfying the identity x^n=0. Under assumption that the characteristic p of the base field is greater than n/2, it is shown that…

We consider the variety of nilpotent elements in the dual of the Lie algebra of a reductive algebraic group over an algebraically closed field. We propose a definition of a partition of this variety into smooth locally closed smooth…

In this paper we continue the study of powerfully nilpotent groups. These are powerful $p$-groups possessing a central series of a special kind. To each such group one can attach a powerful nilpotency class that leads naturally to the…

Let $G$ be a finite soluble group and $G^{(k)}$ the $k$th term of the derived series of $G$. We prove that $G^{(k)}$ is nilpotent if and only if $|ab|=|a||b|$ for any $\delta_k$-values $a,b\in G$ of coprime orders. In the course of the…

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

The aim of this work is to find some exact sequences on the $c$- nilpotent multiplier of a group $G$. We also give an upper bound for the $c$- nilpotent multiplier of finite $p$-groups and give the explicit structure of groups whose take…

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