Minimal and random generation of permutation and matrix groups

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We prove explicit bounds on the numbers of elements needed to generate various types of finite permutation groups and finite completely reducible matrix groups, and present examples to show that they are sharp in all cases. The bounds are linear in the degree of the permutation or matrix group in general, and logarithmic when the group is primitive. They can be combined with results of Lubotzky to produce explicit bounds on the number of random elements required to generate these groups with a specified probability. These results have important applications to computational group theory. Our proofs are inductive and largely theoretical, but we use computer calculations to establish the bounds in a number of specific small cases.
Original languageEnglish
Pages (from-to)195-214
Number of pages20
JournalJournal of Algebra
Early online date12 May 2013
Publication statusPublished - 1 Aug 2013


  • Finite groups
  • Generation of groups
  • Matrix groups
  • Permutation groups
  • Probabilistic and symptotic group theory


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