Invariable generation of permutation and linear groups

Gareth Tracey

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A subset {x 1,x 2,…,x d} of a group G invariably generates G if {x 1 g 1 ,x 2 g 2 ,…,x d g d } generates G for every d-tuple (g 1,g 2…,g d)∈G d. We prove that a finite completely reducible linear group of dimension n can be invariably generated by ⌊[Formula presented]⌋ elements. We also prove tighter bounds when the field in question has order 2 or 3. Finally, we prove that a transitive [respectively primitive] permutation group of degree n≥2 [resp. n≥3] can be invariably generated by O([Formula presented]) [resp. O([Formula presented])] elements.

Original languageEnglish
Pages (from-to)250-289
Number of pages40
JournalJournal of Algebra
Early online date4 Feb 2019
Publication statusPublished - 15 Apr 2019


  • Conjugation properties in finite groups
  • Generation of finite groups

ASJC Scopus subject areas

  • Algebra and Number Theory


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