Carmine Monetta

University of Salerno | UNISA · Department of Mathematics DIPMAT

Let G be a finite group of order n, and denote by ρ(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho (G)$$\end{document} the product of element orders of G. The a...

Let $\Gamma_G$ denote a graph associated with a group $G$. A compelling question about finite groups asks whether or not a finite group $H$ must be nilpotent provided $\Gamma_H$ is isomorphic to $\Gamma_G$ for a finite nilpotent group $G$. In the present work we analyze the problem for different graphs that one can associate with a finite group, bo...

The solubility graph ΓS(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _S(G)$$\end{document} associated with a finite group G is a simple graph whose vertices...

In this work we discuss whether the non-commuting graph of a finite group can determine its nilpotency. More precisely, Abdollahi, Akbari and Maimani conjectured that if $G$ and $H$ are finite groups with isomorphic non-commuting graphs and $G$ is nilpotent, then $H$ must be nilpotent as well (Conjecture 2). We pose a new conjecture (Conjecture 3)...

In this paper we consider two functions related to the arithmetic and geometric means of element orders of a finite group, showing that certain lower bounds on such functions strongly affect the group structure. In particular, for every prime p, we prove a sufficient condition for a finite group to be p-nilpotent, that is, a group whose elements of...

Let G be a finite p‐group. In this paper we obtain bounds for the exponent of the non‐abelian tensor square G⊗G$G \otimes G$ and a certain extension ν(G)$\nu (G)$ of G⊗G$G \otimes G$ by G×G$G \times G$. In particular, we bound exp(ν(G))$\exp (\nu (G))$ in terms of exp(ν(G/N))$\exp (\nu (G/N))$ and exp(N)$\exp (N)$ when G admits some specific normal...

In this paper we address the problem of understanding when a verbal subgroup of a finite group is p-nilpotent, with p a prime, that is, when all its elements of p′-order determine a subgroup. We provide two p-nilpotency criteria, one for the terms of the lower central series of any finite group and one for the terms of the derived series of any fin...

Let $G$ be a finite group of order $n$, and denote by $\rho(G)$ the product of element orders of $G$. The aim of this work is to provide some upper bounds for $\rho(G)$ depending only on $n$ and on its least prime divisor, when $G$ belongs to some classes of non-cyclic groups.

The solubility graph $\Gamma_S(G)$ associated with a finite group $G$ is a simple graph whose vertices are the elements of $G$, and there is an edge between two distinct vertices if and only if they generate a soluble subgroup. In this paper, we focus on the set of neighbors of a vertex $x$ which we call it the solubilizer of $x$ in G, investigatin...

Let G be a group. We denote by ν(G) a certain extension of the non-abelian tensor square G⊗G by G×G. In this paper we prove that the derived subgroup ν(G)′ is a central product of three normal subgroups of ν(G), all isomorphic to the non-abelian tensor square G⊗G. As a consequence, we describe the structure of each term of the derived and lower cen...

Let $G$ be a finite group, let $p$ be a prime and let $w$ be a group-word. We say that $G$ satisfies $P(w,p)$ if the prime $p$ divides the order of $xy$ for every $w$-value $x$ in $G$ of $p'$-order and for every non-trivial $w$-value $y$ in $G$ of order divisible by $p$. If $k \geq 2$, we prove that the $k$th term of the lower central series of $G$...

Let $G$ be a group. We denote by $\nu(G)$ a certain extension of the non-abelian tensor square $G \otimes G$ by $G \times G$. In this paper we prove that the derived subgroup $\nu(G)'$ is a central product of three normal subgroups of $\nu(G)$, all isomorphic to the non-abelian tensor square $G \otimes G$. As a consequence, we describe the structur...

Let $G$ be a group. We denote by $\nu(G)$ a certain extension of the non-abelian tensor square $[G,G^{\varphi}]$ by $G \times G$. In this paper we obtain bounds for the exponent of $\nu(G)$, when $G$ is a finite $p$-group. In particular, we prove that if $N$ is a potent normal subgroup of a $G$, then $\exp(\nu(G))$ divides ${\bf p} \cdot \exp(N) \c...

A subgroup H of a group G is said to be complemented in G if there exists a subgroup K of G such that \(G=HK\) and \(H \cap K=1\). We prove that, for a locally soluble group G, all cyclic subgroups are complemented if and only if it is the semidirect product of groups \(A= {{\,\mathrm{Dr}\,}}_{i \in I} A_i\) by \(B={{\,\mathrm{Dr}\,}}_{j \in J} B_j...

Let $n$ be a positive integer and let $G$ be a group. We denote by $\nu(G)$ a certain extension of the non-abelian tensor square $G \otimes G$ by $G \times G$. Set $T_{\otimes}(G) = {g \otimes h | g, h \in G}$. We prove that if the size of the conjugacy class $|x^{nu(G)| \leqslant n$ for every $x \in T_{\otimes}(G)$, then the second derived subgrou...

Let G be a finite group and let k≥2. We prove that the coprime subgroup γk*(G) is nilpotent if and only if |xy|=|x||y| for any γk*-commutators x,y∈G of coprime orders (Theorem A). Moreover, we show that the coprime subgroup δk*(G) is nilpotent if and only if |ab|=|a||b| for any powers of δk*-commutators a,b∈G of coprime orders (Theorem B).

The word $w=[x_{i_{1}},x_{i_{2}},\ldots ,x_{i_{k}}]$ is a simple commutator word if $k\geq 2,i_{1}\neq i_{2}$ and $i_{j}\in \{1,\ldots ,m\}$ for some $m>1$ . For a finite group $G$ , we prove that if $i_{1}\neq i_{j}$ for every $j\neq 1$ , then the verbal subgroup corresponding to $w$ is nilpotent if and only if $|ab|=|a||b|$ for any $w$ -values $a...

The word $w=[x_{i_1},x_{i_2},\dots,x_{i_k}]$ is a simple commutator word if $k\geq 2, i_1\neq i_2$ and $i_j\in \{1,\dots,m\}$, for some $m>1$. For a finite group $G$, we prove that if $i_{1} \neq i_j$ for every $j\neq 1$, then the verbal subgroup corresponding to $w$ is nilpotent if and only if $|ab|=|a||b|$ for any $w$-values $a,b\in G$ of coprime...

Let $G$ be a finite group and let $k \geq 2$. We prove that the coprime subgroup $\gamma_k^*(G)$ is nilpotent if and only if $|xy|=|x||y|$ for any $\gamma_k^*$-commutators $x,y \in G$ of coprime orders (Theorem A). Moreover, we show that the coprime subgroup $\delta_k^*(G)$ is nilpotent if and only if $|ab|=|a||b|$ for any powers of $\delta_k^*$-co...

We prove that the $k$th term of the lower central series of a finite group $G$ is nilpotent if and only if $|ab| = |a||b|$ for any $gamma_k$-commutators $a,b \in G$ of coprime orders.