On the Normalizer-Solubilizer Conjecture_V3

TL;DR

This paper investigates the normalizer-solubilizer conjecture in finite groups, providing an equivalent condition, verifying it in specific cases involving Frobenius groups, and classifying certain simple groups with maximal solubilizer subgroups.

Contribution

It offers an equivalent formulation of the conjecture, proves it for Frobenius group cases, and classifies simple groups with maximal solubilizer subgroups of order pq.

Findings

Equivalent condition for the normalizer-solubilizer conjecture.

Verification of the conjecture when the normalizer is a Frobenius group.

Classification of simple groups with solubilizer subgroups of order pq.

Abstract

Let $G$ be a finite group and $x$ be an element of $G$. Define $\textrm{Sol}_G(x)$ as the set of all $y \in G$ such that $\langle {x,y}\rangle$ is soluble. We provide an equivalent condition for the normalizer-solubilizer conjecture, namely $|\mathcal{N}_G(\langle x\rangle)| \mid |\textrm{Sol}_G(x)|$, where $\mathcal{N}_G(\langle x\rangle)$ is the normalizer of $\langle x\rangle$. Furthermore, we demonstrate that the conjecture holds in the special case where $\mathcal{N}_G(\langle x\rangle)$ is a Frobenius group with kernel $\mathcal{C}_G(x)$, the centralizer of $x$, and $|\mathcal{N}_G(\langle x\rangle): \mathcal{C}_G(x)|$ is of prime order. Finally, we will classify all finite simple groups $G$ that contain an element $x$ for which $\textrm{Sol}_G(x)$ is a maximal subgroup of order $pq$, where $p$ and $q$ are prime numbers.