Probabilistic Generation of Finite Almost Simple Groups
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929288637317120 |
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| author | Fulman, Jason Garzoni, Daniele Guralnick, Robert M. |
| author_facet | Fulman, Jason Garzoni, Daniele Guralnick, Robert M. |
| contents | We prove that if G is a sufficiently large finite almost simple group of Lie type, then given a fixed nontrivial element x in G and a coset of G modulo its socle, the probability that x and a random element of the coset generate a subgroup containing the socle is uniformly bounded away from 0 (and goes to 1 if the field size goes to infinity). This is new even if G is simple. Together with results of Lucchini and Burness--Guralnick--Harper, this proves a conjecture of Lucchini and has an application to profinite groups. A key step in the proof is the determination of the limits for the proportion of elements in a classical group which fix no subspace of any bounded dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_17291 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Probabilistic Generation of Finite Almost Simple Groups Fulman, Jason Garzoni, Daniele Guralnick, Robert M. Group Theory Combinatorics We prove that if G is a sufficiently large finite almost simple group of Lie type, then given a fixed nontrivial element x in G and a coset of G modulo its socle, the probability that x and a random element of the coset generate a subgroup containing the socle is uniformly bounded away from 0 (and goes to 1 if the field size goes to infinity). This is new even if G is simple. Together with results of Lucchini and Burness--Guralnick--Harper, this proves a conjecture of Lucchini and has an application to profinite groups. A key step in the proof is the determination of the limits for the proportion of elements in a classical group which fix no subspace of any bounded dimension. |
| title | Probabilistic Generation of Finite Almost Simple Groups |
| topic | Group Theory Combinatorics |
| url | https://arxiv.org/abs/2403.17291 |