## Abstract

Let G be a finite classical group generated by transvections, i.e., one of SL

By combining this with a result of the author and Jezernikit follows that if G is one of SL

_{n}(q), SU_{n}(q), Sp_{2n}(q), or O^{±}_{2n}(q) (q even),and let X be a generating set for G containing at least one transvection. Building on work of Garonzi, Halasi, and Somlai, we prove that the diameter of the Cayley graph Cay(G, X) is bounded by (n log q)^{C}for some constant C. This confirms Babai’s conjecture on the diameter of finite simple groups in the case of generating sets containing a transvection.By combining this with a result of the author and Jezernikit follows that if G is one of SL

_{n}(q), SU_{n}(q), Sp_{2n}(q) and X contains three random generators then with high probability the diameter Cay(G, X) is bounded by n^{O(log q)}. This confirms Babai’s conjecture for non-orthogonal classical simple groups over small fields and three random generators.Original language | English |
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Pages (from-to) | 220-256 |

Number of pages | 37 |

Journal | Journal of Algebra |

Volume | 653 |

Early online date | 20 May 2024 |

DOIs | |

Publication status | Early online date - 20 May 2024 |

## Keywords

- Babai's conjecture
- Classical groups
- Diameter
- Transvections

## ASJC Scopus subject areas

- Algebra and Number Theory