Derangements in finite classical groups and characteristic polynomials of random matrices
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912975648980992 |
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| author | Fulman, Jason Guralnick, Robert |
| author_facet | Fulman, Jason Guralnick, Robert |
| contents | We first obtain explicit upper bounds for the proportion of elements in a finite classical group G with a given characteristic polynomial. We use this to complete the proof that the proportion of elements of a finite classical group G which lie in a proper irreducible subgroup tends to 0 as the dimension of the natural module goes to infinity. This result is analogous to the result of Luczak and Pyber [15] that the proportion of elements of the symmetric group S_n which are contained in a proper transitive subgroup other than the alternating group goes to 0 as n goes to infinity. We also show that the probability that 3 random elements of SL(n,q) invariably generate goes to 0 as n goes to infinity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_21025 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Derangements in finite classical groups and characteristic polynomials of random matrices Fulman, Jason Guralnick, Robert Group Theory Combinatorics Probability We first obtain explicit upper bounds for the proportion of elements in a finite classical group G with a given characteristic polynomial. We use this to complete the proof that the proportion of elements of a finite classical group G which lie in a proper irreducible subgroup tends to 0 as the dimension of the natural module goes to infinity. This result is analogous to the result of Luczak and Pyber [15] that the proportion of elements of the symmetric group S_n which are contained in a proper transitive subgroup other than the alternating group goes to 0 as n goes to infinity. We also show that the probability that 3 random elements of SL(n,q) invariably generate goes to 0 as n goes to infinity. |
| title | Derangements in finite classical groups and characteristic polynomials of random matrices |
| topic | Group Theory Combinatorics Probability |
| url | https://arxiv.org/abs/2507.21025 |