A note on the probability of generating alternating or symmetric groups


We improve on recent estimates for the probability of generating the alternating and symmetric groups A n and S n . In particular, we find the sharp lower bound if the probability is given by a quadratic in n −1. This leads to improved bounds on the largest number h(A n ) such that a direct product of h(A n ) copies of A n can be generated by two elements.

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Corresponding author

Correspondence to Luke Morgan.

Additional information

The research of the first author is supported by the Australian Research Council grant DP120100446. This work was done whilst the second author was visiting The University of Western Australia as a Cheryl E. Praeger Visiting Research Fellow. The authors would like to thank the anonymous referee for their extremely helpful remarks.

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Morgan, L., Roney-Dougal, C.M. A note on the probability of generating alternating or symmetric groups. Arch. Math. 105, 201–204 (2015). https://doi.org/10.1007/s00013-015-0796-8

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Mathematics Subject Classification

  • 20B30
  • 20P05


  • Symmetric group
  • Alternating group
  • Generation
  • Probability