Searcharxiv⌕ Search

arXiv subjects

Costantino Delizia

Publications and source records attributed to Costantino Delizia.

10 recordsLinked to original sources
arXiv2025

On the directed normalizing graph associated with a group

In this paper we investigate the $directed$ $normalizing$ $graph$ associated with a group $G$, defined as the simple directed graph whose vertices are the elements of $G$, with an arrow from $x$ to $y$ whenever the subgroup $\langle x \rangle$ is normal in $\langle x, y \rangle$. Our analysis focuses on the set of bidirectional universal vertices and, in particular, on the induced subgraph obtained by removing them, where the most interesting connectivity phenomena occur. We characterize the groups for which this induced subgraph is strongly connected and determine bounds for its diameter. Finally, we show how properties of this graph reflect algebraic features of the underlying group.

math.GR↗
arXiv2025

Neighborhoods, connectivity, and diameter of the nilpotent graph of a finite group

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 the nilpotent graph of a finite group $G$. Indeed, we characterize finite solvable groups whose closed neighborhoods are nilpotent subgroups. Moreover, we study the connectivity of the graph $Γ(G)$ obtained removing all universal vertices from the nilpotent graph of $G$. Some upper bounds to the diameter of $Γ(G)$ are provided when $G$ belongs to some classes of groups.

math.GR↗
arXiv2024

A generalization of concise words

The study of verbal subgroups within a group is well-known for being an effective tool to obtain structural information about a group. Therefore, conditions that allow the classification of words in a free group are of paramount importance. One of the most studied problems is to establish which words are concise, where a word $w$ is said to be concise if the verbal subgroup $w(G)$ is finite in each group $G$ in which $w$ takes only a finite number of values. The purpose of this article is to present some results, in which a hierarchy among words is introduced, generalizing the concept of concise word.

math.GR↗
arXiv2022

On the solubilizer of an element in a finite group

The solubility graph $Γ_S(G)$ 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 the solubilizer of $x$ in $G$, $\mathrm{Sol}_G(x)$, investigating both arithmetic and structural properties of this set.

math.GR↗
arXiv2016

Groups in which every non-abelian subgroup is self-normalized

We study groups having the property that every non-abelian subgroup is equal to its normalizer. This class of groups is closely related to an open problem posed by Berkovich. We give a full classification of finite groups having the above property. We also describe all infinite soluble groups in this class.

math.GR↗
arXiv2016

Groups in which every non-abelian subgroup is self-centralizing

We study groups having the property that every non-abelian subgroup contains its centralizer. We describe various classes of infinite groups in this class, and address a problem of Berkovich regarding the classification of finite $p$-groups with the above property.

math.GR↗
arXiv2013

Groups in which every non-cyclic subgroup contains its centralizer

We study groups having the property that every non-cyclic subgroup contains its centralizer. The structure of nilpotent and supersolvable groups in this class is described. We also classify finite $p$-groups and finite simple groups with the above defined property.

math.GR↗