Generation and random generation: From simple groups to maximal subgroups

Timothy C. Burness*, Martin W. Liebeck, Aner Shalev

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


Let G be a finite group and let d( G) be the minimal number of generators for G. It is well known that d( G) = 2 for all (non-abelian) finite simple groups. We prove that d( H) ≤ 4 for any maximal subgroup H of a finite simple group, and that this bound is best possible.We also investigate the random generation of maximal subgroups of simple and almost simple groups. By applying a recent theorem of Jaikin-Zapirain and Pyber we show that the expected number of random elements generating such a subgroup is bounded by an absolute constant.We then apply our results to the study of permutation groups. In particular we show that if G is a finite primitive permutation group with point stabilizer H, then d( G) - 1 ≤ d( H) ≤ d( G) + 4.

Original languageAmerican English
Pages (from-to)59-95
Number of pages37
JournalAdvances in Mathematics
StatePublished - 5 Nov 2013

Bibliographical note

Funding Information:
The first author acknowledges the support of EPSRC grant EP/I019545/1 . The second and third authors acknowledge the support of EPSRC grant EP/H018891/1 . The third author also acknowledges the support of an Advanced ERC Grant, an Israel Science Foundation Grant, and the Miriam and Julius Vinik Chair in Mathematics which he holds.


  • Finite simple groups
  • Maximal subgroups
  • Minimal generation
  • Primitive permutation groups


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