The probability of generating a finite simple group

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We study the probability of generating a finite simple group, together with its generalisation PG,socG(d), the conditional probability of generating an almost simple finite group G by d elements, given that these elements generate G/ socG. We prove that PG,socG(2) ⩾ 53/90, with equality if and only if G is A6 or S6, and establish a similar result for PG,socG(3). Positive answers to longstanding questions of Wiegold on direct products, and of Mel’nikov on profinite groups, follow easily from our results.
Original languageEnglish
Pages (from-to)371-392
Number of pages22
JournalIsrael Journal of Mathematics
Issue number1
Early online date1 Jul 2013
Publication statusPublished - Nov 2013


  • Probability
  • Finite simple group
  • Direct products
  • Profinite groups


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