Characterization of Solubilizers of Elements in Minimal Simple Groups

TL;DR

This paper classifies solubilizers of elements in minimal simple groups, exploring their properties and proposing computational methods and conjectures for future research.

Contribution

It provides the first comprehensive classification of solubilizers in minimal simple groups and investigates their properties with computational approaches.

Findings

Classified all solubilizers in minimal simple groups.

Identified key properties of solubilizer sets.

Proposed computational methods and conjectures for further study.

Abstract

Given a finite group $G$, the solubilizer of an element $x$, denoted by $\Sol_G(x)$, is the set of all elements $y$ such that $\langle x, y\rangle$ is a soluble subgroup of $G$. In this paper, we provide a classification for all solubilizers of elements in minimal simple groups. We also examine these sets to explore their properties by discussing some computational methods and making some conjectures for further work.