On the conjugacy class exponent of the finite simple groups
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915415966351360 |
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| author | Garonzi, Martino Montijo, Christe Zalesski, Alexandre |
| author_facet | Garonzi, Martino Montijo, Christe Zalesski, Alexandre |
| contents | The generalized order $e_G(g)$ of an element $g$ of a group $G$ is the smallest positive integer $k$ such that there exist $x_1,\ldots,x_k \in G$ such that $g^{x_1} \ldots g^{x_k}=1$, where $g^x=x^{-1}gx$. Let $e(G) = \max \{e_G(g)\ |\ g \in G\}$. We provide upper bounds for $e(G)$ for every finite simple group $G$. In particular, we show that $e(G)\leq 8$ unless $G\in\{\mbox{PSL}_n(q), \mbox{PSU}_n(q), E_6(q),{^2}E_6(q)\}$. For the latter groups $e(G)\leq n,3n+3,36,36$, respectively. In addition, we bound from above the generalized order of semisimple and unipotent elements of finite simple groups of Lie type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_22268 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the conjugacy class exponent of the finite simple groups Garonzi, Martino Montijo, Christe Zalesski, Alexandre Group Theory The generalized order $e_G(g)$ of an element $g$ of a group $G$ is the smallest positive integer $k$ such that there exist $x_1,\ldots,x_k \in G$ such that $g^{x_1} \ldots g^{x_k}=1$, where $g^x=x^{-1}gx$. Let $e(G) = \max \{e_G(g)\ |\ g \in G\}$. We provide upper bounds for $e(G)$ for every finite simple group $G$. In particular, we show that $e(G)\leq 8$ unless $G\in\{\mbox{PSL}_n(q), \mbox{PSU}_n(q), E_6(q),{^2}E_6(q)\}$. For the latter groups $e(G)\leq n,3n+3,36,36$, respectively. In addition, we bound from above the generalized order of semisimple and unipotent elements of finite simple groups of Lie type. |
| title | On the conjugacy class exponent of the finite simple groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2506.22268 |