Abstract
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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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). https://doi.org/10.1007/BF01271271
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DOI: https://doi.org/10.1007/BF01271271