On the solubilizer of an element in a finite group

TL;DR

This paper studies the solubilizer of an element in a finite group, analyzing its properties within the context of the solubility graph, which encodes soluble subgroup generation.

Contribution

It introduces and investigates the properties of solubilizers of elements in finite groups through the solubility graph framework.

Findings

Characterization of solubilizers in finite groups

Structural properties of solubilizer sets

Insights into the solubility graph structure

Abstract

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