The impact of the solubilizer of an element on the structure of a finite group

TL;DR

This paper investigates how the set of elements that generate soluble subgroups with a fixed element influences the overall structure of finite groups.

Contribution

It introduces the concept of $ ext{Sol}_G(x)$ and explores its impact on the structural properties of finite groups, providing new insights into group solubility relations.

Findings

$ ext{Sol}_G(x)$ characterizes subgroup structures

Influence of $ ext{Sol}_G(x)$ on group solubility

Structural constraints derived from $ ext{Sol}_G(x)$

Abstract

Let $G$ be a finite group, and let $x$ be an element of $G$. Denote by $\Sol_G(x)$ the set of all $y \in G$ such that the group generated by $x$ and $y$ is soluble. We investigate the influence of $\Sol_G(x)$ on the structure of $G$.