Related papers: Structure of the Macdonald groups in one parameter

For an element $g$ of a group $G$, an Engel sink is a subset ${\mathscr E}(g)$ such that for every $x\in G$ all sufficiently long commutators $[...[[x,g],g],\dots ,g]$ belong to ${\mathscr E}(g)$. A~finite group is nilpotent if and only if…

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…

We study finitely generated nilpotent groups $G$ given by full rank finite presentations $\langle A \mid R\rangle$ in the variety $\mathcal{N}_c$ of nilpotent groups of class at most $c$, where $c \geq 2$. We prove that if the deficiency…

Suppose that a finite group $G$ admits a Frobenius group of automorphisms $FH$ with kernel $F$ and complement $H$ such that the fixed-point subgroup of $F$ is trivial: $C_G(F)=1$. In this situation various properties of $G$ are shown to be…

A finite group $G$ is said to have the nilpotent decomposition property (ND) if for every nilpotent element $\alpha$ of the integral group ring $\mathbb{Z}[G]$ one has that $\alpha e$ also belong to $\mathbb{Z}[G]$, for every primitive…

In this article we show that if ${\cal V}$ is the variety of polynilpotent groups of class row $(c_1,c_2,...,c_s),\ {\mathcal N}_{c_1,c_2,...,c_s}$, and $G\cong{\bf {Z}}_{p^{\alpha_1}}\stackrel{n}{*}{\bf…

We introduce a model of random f.g., torsion-free, $2$-step nilpotent groups (in short, $\tau_2$-groups). To do so, we show that these are precisely the groups that admit a presentation of the form $ \label{tau2pres_0}\langle A, C \mid…

For any group G, let C(G) denote the intersection of the normal- izers of centralizers of all elements of G. Set C0 = 1. Define Ci+1(G)=Ci(G) = C(G=Ci(G)) for i ? 0. By C1(G) denote the terminal term of the ascending series. In this paper,…

Let $F$ be a nilpotent group acted on by a group $H$ via automorphisms and let the group $G$ admit the semidirect product $FH$ as a group of automorphisms so that $C_G(F) = 1$. We prove that the order of $\gamma_\infty(G)$, the rank of…

Suppose $G$ is a $\mathcal{T}$-group (finitely generated torsion-free nilpotent) with centralizers outside of the derived subgroup being abelian of rank equal to $\text{rank}(Z_1)+1$. This includes the class of free nilpotent groups…

In studying nilpotent groups, the lower central series and other variations can be used to construct an associated $\mathbb{Z}^+$-graded Lie ring, which is a powerful method to inspect a group. Indeed, the process can be generalized…

Let $G$ be a finite group and $H$ be a subgroup of $G$. Then $H$ is said to be a $p$-$CAP$-subgroup of $G$, if $H$ covers or avoids any $pd$-chief factor of $G$. Furthermore, $H$ is said to be a strong $p$-$CAP$-subgroup of $G$, if for any…

Let $n>0$ be an integer and $\mathcal{X}$ be a class of groups. We say that a group $G$ satisfies the condition $(\mathcal{X},n)$ whenever in every subset with $n+1$ elements of $G$ there exist distinct elements $x,y$ such that $<x,y>$ is…

We prove several results detecting ciclicity or nilpotency of a finite group $G$ in terms of inequalities involving the orders of the elements of $G$ and the orders of the elements of the cyclic group of order $|G|$. We prove that, among…

Let $G$ be a nilpotent group and $a\in G$. Let $a^G=\{g^{-1}ag\mid g\in G\}$ be the conjugacy class of $a$ in $G$. Assume that $a^G$ and $b^G$ are conjugacy classes of $G$ with the property that $|a^G|=|b^G|=p$, where $p$ is an odd prime…

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…

In this paper, we determine the structure of the nilpotent multipliers of all pairs $(G,N)$ of finitely generated abelian groups where $N$ admits a complement in $G$. Moreover, some inequalities for the nilpotent multipliers of pairs of…

In this paper, several theorems of Macdonald \cite{Mac1961,Mac1962} on the varieties of nilpotent groups will be generalized to the case of Lie rings. We consider three varieties of Lie rings of any characteristic associated with some…

In Part I it was shown that if G is a p-group of class k, generated by elements of orders 1<p^{alpha_1} <= ... <= p^{alpha_r}, then a necessary condition for the capability of G is that r>1 and alpha_r <= alpha_{r-1} + [(k-1)/(p-1)]. It was…

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…