On the Normalizer-Solubilizer Conjecture_V3

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.