On the proportion of elements of prime order in finite symmetric groups

Document Type : 2022 CCGTA IN SOUTH FLA


1 Centre for the Mathematics of Symmetry and Computation, University of Western Australia, 35 Stirling Highway, Perth 6009, Australia

2 Department of Mathematics, Federal University Gashua, Yobe State, Nigeria


We give a short proof for an explicit upper bound on the proportion of permutations of a given prime order $p$, acting on a set of given size $n$, which is sharp for certain $n$ and $p$. Namely, we prove that if $n\equiv k\pmod{p}$ with $0\leq k\leq p-1$, then this proportion is at most $(p\cdot k!)^{-1}$ with equality if and only if $p\leq n<2n$.


Main Subjects

[1] J. Bamberg, S. P. Glasby, S. Harper and C. E. Praeger, Permutations with orders coprime to a given integer,
Electronic J. Combin., 27 (2020) 14 pp.
[2] R. Beals, Charles R. Leedham-Green, A. C. Niemeyer, C. E. Praeger and Á. Seress, Permutations with restricted
cycle structure and an algorithmic application, Combin. Probab. Comput., 11 (2002) 447–464.
[3] S. Chowla, I. N. Herstein and W. R. Scott, The solutions of xd = 1 in symmetric groups, Norske Vid. Selsk. Forh.,
Trondheim, 25 (1952) 29–31.
[4] S. P. Glasby, C. E. Praeger and W. R. Unger, Most permutations power to a cycle of small prime length, Proc.
Edinb. Math. Soc. (2), 64 (2021) 234–246.
[5] E. Jacobsthal, Sur le nombre d’éléments du groupe symétrique Sn dont l’ordre est un nombre premier, Norske Vid. Selsk. Forh., Trondheim, 21 (1949) 49–51.
[6] F. Lübeck, A. C. Niemeyer and C. E. Praeger, Finding involutions in finite Lie type groups of odd characteristic, J.
Algebra, 321 (2009) 3397–3417.
[7] L. Moser and M. Wyman, On solutions of xd = 1 in symmetric groups, Canadian J. Math., 7 (1955) 159–168.
[8] A. C. Niemeyer, T. Popiel and C. E. Praeger, On proportions of pre-involutions in finite classical groups, J. Algebra,
324 (2010) 1016–1043.
[9] A. C. Niemeyer, C. E. Praeger and Á. Seress, Estimation problems and randomised group algorithms, In Probabilistic Group Theory, Combinatorics and Computing, Editors: Alla Detinko, Dane Flannery and Eamonn O’Brien, Lecture
Notes in Mathematics, Springer, Berlin, 2070 2020 35–82.
[10] H. S. Wilf, The asymptotics of eP (z) and the number of elements of each order in Sn , Bull. Amer. Math. Soc. (N.S.), 15 (1986) 228–232.
Volume 13, Issue 3 - Serial Number 3
September 2024
Pages 251-256
  • Receive Date: 25 October 2022
  • Revise Date: 06 April 2023
  • Accept Date: 08 April 2023
  • Published Online: 01 September 2024