On the normalizer-solubilizer conjecture

Mousavi, Hamid

doi:10.22108/ijgt.2025.143871.1938

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	On the normalizer-solubilizer conjecture
International Journal of Group Theory
دوره 15، شماره 3، آذر 2026، صفحه 107-122 اصل مقاله (491.53 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/ijgt.2025.143871.1938
نویسنده
Hamid Mousavi^*
Department of Mathematics, University of Tabriz, P.O.Box 51666-17766, Tabriz, Iran
چکیده
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.
کلیدواژه‌ها
Finite simple group؛ Insoluble group؛ Solubilizer

مراجع
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On the normalizer-solubilizer conjecture