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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2410.16931 |
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| _version_ | 1866909359025422336 |
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| author | Degos, Jean-Yves |
| author_facet | Degos, Jean-Yves |
| contents | Let $p$ be a primer number, $n \geq 3$ and integer. Let $f(X) = X^n + a_{n-1}X^{n-1} + \cdots +a_1 X + a_0 \in \mathbb{F}_p[X]$ be a primitive polynomial of degree $n$. Let $C_f$ be the companion matrix of $f(X)$, and $G$ the companion matrix of the polynomial $X^n-1$. Define $G_1 := C_f$ and $G_{k+1} = G G_k G^{-1}$ for $0 \leq k \leq n-1$. The so called ``Brunnian Conjecture'' states that: the general linear group $GL(n,p)$ is generated by $G_1, G_2, \ldots, G_n$. In this paper, we prove it for $p \geq 5$ and $n$ not divisible by $p-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16931 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Brunnian conjecture Degos, Jean-Yves Group Theory Let $p$ be a primer number, $n \geq 3$ and integer. Let $f(X) = X^n + a_{n-1}X^{n-1} + \cdots +a_1 X + a_0 \in \mathbb{F}_p[X]$ be a primitive polynomial of degree $n$. Let $C_f$ be the companion matrix of $f(X)$, and $G$ the companion matrix of the polynomial $X^n-1$. Define $G_1 := C_f$ and $G_{k+1} = G G_k G^{-1}$ for $0 \leq k \leq n-1$. The so called ``Brunnian Conjecture'' states that: the general linear group $GL(n,p)$ is generated by $G_1, G_2, \ldots, G_n$. In this paper, we prove it for $p \geq 5$ and $n$ not divisible by $p-1$. |
| title | On the Brunnian conjecture |
| topic | Group Theory |
| url | https://arxiv.org/abs/2410.16931 |