arXiv subjects
Publications and source records attributed to Nicolas Pinzauti.
We define a class of finite groups based on the properties of the closed twins of their power graphs and study the structure of those groups. As a byproduct, we obtain results about finite groups admitting a partition by cyclic subgroups.
In a graph $\Gamma=(V,E)$, we consider the common closed neighbourhood of a subset of vertices and use this notion to introduce a Moore closure operator in $V.$ We also consider the closed twin equivalence relation in which two vertices are equivalent if they have the same closed neighbourhood. Those notions are deeply explored when $\Gamma$ is the power graph associated with a finite group $G$. In that case, among the corresponding closed twin equivalence classes, we introduce the concepts of plain, compound and critical classes. The study of critical classes, together with properties of the Moore closure operator, allow us to correct a mistake in the proof of {\rm \cite[Theorem 2 ]{Cameron_2}} and to deduce a simple algorithm to reconstruct the directed power graph of a finite group from its undirected counterpart, as asked in \cite[Question 2]{GraphsOnGroups}.
The power graph $\mathcal{P}(G)$ of a group $G$ is the graph whose vertex set is $G$, having an edge between two distinct vertices if one is the power of the other. The directed power graph $\vec{\mathcal{P}}(G)$ of a group $G$ is the digraph whose vertex set is $G$, having an arc from $x$ to $y$, with $x\ne y$, whenever $y$ is a power of $x$. We rewrite two Cameron's articles concerning the reconstruction of $\vec{\mathcal{P}}(G)$ from $\mathcal{P}(G)$. We correct mistakes that appear in the papers. In particular, we add missing cases needed to complete the main theorems of these articles. We also study the quotient of the power graph under some equivalence relations. We close the thesis with lower bounds for the maximum length of a cycle in the power graph of a group.