University of Isfahan International Journal of Group Theory 2251-7650 15 3 2025 07 08 On the normalizer-solubilizer conjecture 107 122 29630 10.22108/ijgt.2025.143871.1938 EN Hamid Mousavi Department of Mathematics, University of Tabriz, P.O.Box 51666-17766, Tabriz, Iran Journal Article 2025 01 01 Let $G$ be a finite group and $x$ be an element of $G$. Define ${\rm Sol}_G(x)$ 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 $|\mathcal{N}_G(\langle{x}\rangle)| \mid |{\rm Sol}_G(x)|$, 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 ${\rm Sol}_G(x)$ is a maximal subgroup of order $pq$, where $p$ and $q$ are prime numbers. https://ijgt.ui.ac.ir/article_29630_9bcf8863ef5426a3f803f018838306df.pdf