On the Normalizer-Solubilizer Conjecture_

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