Diameter of classical groups generated by transvections

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Let G be a finite classical group generated by transvections, i.e., one of SLn(q), SUn(q), Sp2n(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 SLn(q), SUn(q), Sp2n(q) and X contains three random generators then with high probability the diameter Cay(G, X) is bounded by nO(log q). This confirms Babai’s conjecture for non-orthogonal classical simple groups over small fields and three random generators.

Original languageEnglish
Pages (from-to)220-256
Number of pages37
JournalJournal of Algebra
Early online date20 May 2024
Publication statusEarly online date - 20 May 2024


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

ASJC Scopus subject areas

  • Algebra and Number Theory


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