On the Spread of Finite Simple Groups

The spread of a finite group is the maximal integer k so that for each k non-identity elements of G there is an element generating G with each of them. We prove an asymptotic result characterizing the finite simple groups of bounded spread. We also obtain estimates for the spread of the various families of finite simple groups, and show that it is at least 2, with possibly finitely many exceptions. The proofs involve probabilistic methods.

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Correspondence to Robert M. Guralnick.

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The first author acknowledges the support of the NSF; the second author acknowledges the support of the Israel Science Foundation and the hospitality of USC; both authors acknowledge the support and hospitality of MSRI.

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Guralnick, R.M., Shalev, A. On the Spread of Finite Simple Groups. Combinatorica 23, 73–87 (2003). https://doi.org/10.1007/s00493-003-0014-3

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AMS Subject Classification (2000):

  • 20B30
  • 20D08
  • 20P05