This is one of a series of papers that aims to give an explicit upper bound on the proportion of elements of order a product of two primes in finite symmetric groups. This one presents such a bound for the elements as the product of two distinct odd primes.
Citation: Hailin Liu, Longzhi Lu, Liping Zhong. On the proportion of elements of order a product of two primes in finite symmetric groups[J]. AIMS Mathematics, 2024, 9(9): 24394-24400. doi: 10.3934/math.20241188
This is one of a series of papers that aims to give an explicit upper bound on the proportion of elements of order a product of two primes in finite symmetric groups. This one presents such a bound for the elements as the product of two distinct odd primes.
| [1] |
R. Beals, C. R. Leedham-Green, A. C. Niemeyer, C. E. Praeger, A. Seress, Permutations with restricted cycle structure and an algorithmic application, Comb. Probab. Comput., 11 (2002), 447–464. http://doi.org/10.1017/S0963548302005217 doi: 10.1017/S0963548302005217
|
| [2] | S. Chowla, I. N. Herstein, W. R. Scott, The solutions of $x^d = 1$ in symmetric groups, Norske Vid. Selsk. Forh. Trondheim, 25 (1952), 29–31. |
| [3] |
S. P. Glasby, C. E. Praeger, W. R. Unger, Most permutations power to a cycle of small prime length, P. Edinburgh Math. Soc., 64 (2021), 234–246. https://doi.org/10.1017/S0013091521000110 doi: 10.1017/S0013091521000110
|
| [4] | E. Jacobsthal, Sur le nombre d'$\acute{e}$l$\acute{e}$ments du groupe sym$\acute{e}$trique ${\rm{S}}_n$ dont l'ordre est un nombre premier, Norske Vid. Selsk. Forh. Trondheim, 21 (1949), 49–51. |
| [5] | H. L. Liu, L. P. Zhong, On the proportion of elements of order $2p$ in finite symmetric groups, Hacet. J. Math. Stat., 2024. Available from: https://doi.org/10.15672/hujms.1367438 |
| [6] |
L. Moser, On solutions of $x^d = 1$ in symmetric groups, Can. J. Math., 7 (1955), 159–168. https://doi.org/10.4153/CJM-1955-021-8 doi: 10.4153/CJM-1955-021-8
|
| [7] |
A. C. Niemeyer, T. Popiel, C. E. Praeger, On proportions of pre-involutions in finite classical groups, J. Algebra, 324 (2010), 1016–1043. https://doi.org/10.1016/j.jalgebra.2010.03.026 doi: 10.1016/j.jalgebra.2010.03.026
|
| [8] | A. C. Niemeyer, C. E. Praeger, A. Seress, Estimation problems and randomised group algorithms. In Probabilistic Group Theory, Combinatorics and Computing, London: Springer Press, 2013, 35–82. https://doi.org/10.1007/978-1-4471-4814-2_2 |
| [9] |
C. E. Praeger, E. Suleiman, On the proportion of elements of prime order in finite symmetric groups, Int. J. Group Theory, 13 (2024), 251–256. http://dx.doi.org/10.22108/ijgt.2023.135509.1810 doi: 10.22108/ijgt.2023.135509.1810
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Hailin Liu, Longzhi Lu, Liping Zhong. On the proportion of elements of order a product of two primes in finite symmetric groups[J]. AIMS Mathematics, 2024, 9(9): 24394-24400. doi: 10.3934/math.20241188
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