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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Online-Zugang: | https://arxiv.org/abs/2603.22638 |
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| _version_ | 1866915914768711680 |
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| author | Glasby, S. P. Niemeyer, Alice C. Praeger, Cheryl E. |
| author_facet | Glasby, S. P. Niemeyer, Alice C. Praeger, Cheryl E. |
| contents | With high probability, among $O(\log n)$ independent randomly selected elements from a finite $n$-dimensional classical group, some pair of elements power to a $2$-element generating set for a naturally embedded classical subgroup of dimension $O(\log n)$. The $2$-element generating set produced consists of certain elements with large $1$-eigenspaces, called stingray elements. Underpinning this result is a new theorem on the generation of a finite classical group by a pair of stingray elements. In particular we show that, for classical groups not containing ${\rm SL}_n(q)$, the probability of generation is at least $0.975$. The explicit probability bounds we obtain will be applied to justify complexity analyses for new constructive recognition algorithms for finite classical groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_22638 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The probability that two elements with large $1$-eigenspaces generate a classical group Glasby, S. P. Niemeyer, Alice C. Praeger, Cheryl E. Group Theory Representation Theory 20D06, 20P05, 20H20 With high probability, among $O(\log n)$ independent randomly selected elements from a finite $n$-dimensional classical group, some pair of elements power to a $2$-element generating set for a naturally embedded classical subgroup of dimension $O(\log n)$. The $2$-element generating set produced consists of certain elements with large $1$-eigenspaces, called stingray elements. Underpinning this result is a new theorem on the generation of a finite classical group by a pair of stingray elements. In particular we show that, for classical groups not containing ${\rm SL}_n(q)$, the probability of generation is at least $0.975$. The explicit probability bounds we obtain will be applied to justify complexity analyses for new constructive recognition algorithms for finite classical groups. |
| title | The probability that two elements with large $1$-eigenspaces generate a classical group |
| topic | Group Theory Representation Theory 20D06, 20P05, 20H20 |
| url | https://arxiv.org/abs/2603.22638 |