Groups with super-exponential subgroup growth


We show that, if the subgroup growth of a finitely generated (abstract or profinite) groupG is super-exponential, then every finite group occurs as a quotient of a finite index subgroup ofG. The proof involves techniques from finite permutation groups, and depends on the Classification of Finite Simple Groups.

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Additional information

The first author was partially supported by the Hungarian National Foundation for Scientific Research, Grant No. T7441. The second author was partially supported by the Israeli National Science Foundation.

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Pyber, L., Shalev, A. Groups with super-exponential subgroup growth. Combinatorica 16, 527–533 (1996).

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Mathematics Subject Classification (1991)

  • primary: 20E07
  • 20E26
  • secondary: 20B35
  • 20E18