The Chebotarev invariant for direct products of nonabelian finite simple groups
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929469684449280 |
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| author | Anzanello, Jessica Lucchini, Andrea Tracey, Gareth |
| author_facet | Anzanello, Jessica Lucchini, Andrea Tracey, Gareth |
| contents | A subset $\{g_1, \ldots , g_d\}$ of a finite group $G$ invariably generates $G$ if $\{g_1^{x_1}, \ldots , g_d^{x_d}\}$ generates $G$ for every choice of $x_i \in G$. The Chebotarev invariant $C(G)$ of $G$ is the expected value of the random variable $n$ that is minimal subject to the requirement that $n$ randomly chosen elements of $G$ invariably generate $G$. In this paper, we show that if $G$ is a nonabelian finite simple group, then $C(G)$ is absolutely bounded. More generally, we show that if $G$ is a direct product of $k$ nonabelian finite simple groups, then $C(G)=\log{k}/\log{α(G)}+O(1)$, where $α$ is an invariant completely determined by the proportion of derangements of the primitive permutation actions of the factors in $G$. It follows from the proof of the Boston-Shalev conjecture that $C(G)=O(\log{k})$. We also derive sharp bounds on the expected number of generators for $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_12298 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Chebotarev invariant for direct products of nonabelian finite simple groups Anzanello, Jessica Lucchini, Andrea Tracey, Gareth Group Theory A subset $\{g_1, \ldots , g_d\}$ of a finite group $G$ invariably generates $G$ if $\{g_1^{x_1}, \ldots , g_d^{x_d}\}$ generates $G$ for every choice of $x_i \in G$. The Chebotarev invariant $C(G)$ of $G$ is the expected value of the random variable $n$ that is minimal subject to the requirement that $n$ randomly chosen elements of $G$ invariably generate $G$. In this paper, we show that if $G$ is a nonabelian finite simple group, then $C(G)$ is absolutely bounded. More generally, we show that if $G$ is a direct product of $k$ nonabelian finite simple groups, then $C(G)=\log{k}/\log{α(G)}+O(1)$, where $α$ is an invariant completely determined by the proportion of derangements of the primitive permutation actions of the factors in $G$. It follows from the proof of the Boston-Shalev conjecture that $C(G)=O(\log{k})$. We also derive sharp bounds on the expected number of generators for $G$. |
| title | The Chebotarev invariant for direct products of nonabelian finite simple groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2408.12298 |