On the Normalizer-Solubilizer Conjecture_V3
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910997295398912 |
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| author | Mousavi, Hamid |
| author_facet | Mousavi, Hamid |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_11486 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Normalizer-Solubilizer Conjecture_V3 Mousavi, Hamid Group Theory 20D05, 20D99 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. |
| title | On the Normalizer-Solubilizer Conjecture_V3 |
| topic | Group Theory 20D05, 20D99 |
| url | https://arxiv.org/abs/2501.11486 |