V. Sotomayor | Searcharxiv

arXiv2024

Groups whose common divisor graph on $p$-regular classes has diameter three

Let $G$ be a finite $p$-separable group, for some fixed prime $p$. Let $Γ_p(G)$ be the common divisor graph built on the set of non-central conjugacy classes of $p$-regular elements of $G$: this is the graph whose vertices are the conjugacy classes of those non-central elements of $G$ such that $p$ does not divide their orders, and two distinct vertices are adjacent if and only if the greatest common divisor of their lengths is strictly greater than one. The aim of this paper is twofold: to positively answer an open question concerning the maximum possible distance in $Γ_p(G)$ between a vertex with maximal cardinality and any other vertex, and to study the $p$-structure of $G$ when $Γ_p(G)$ has diameter three.

math.GR↗

arXiv2019

On zeros of irreducible characters lying in a normal subgroup

Let $N$ be a normal subgroup of a finite group $G$. In this paper, we consider the elements $g$ of $N$ such that $χ(g)\neq 0$ for all irreducible characters $χ$ of $G$. Such an element is said to be non-vanishing in $G$. Let $p$ be a prime. If all $p$-elements of $N$ satisfy the previous property, then we prove that $N$ has a normal Sylow $p$-subgroup. As a consequence, we also study certain arithmetical properties of the $G$-conjugacy class sizes of the elements of $N$ which are zeros of some irreducible character of $G$. In particular, if $N=G$, then new contributions are obtained.

math.GR↗