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

Praeger, Cheryl E.; Suleiman, Enoch

doi:10.22108/ijgt.2023.135509.1810

تعداد نشریات	43
تعداد شماره‌ها	1,682
تعداد مقالات	13,762
تعداد مشاهده مقاله	32,194,100
تعداد دریافت فایل اصل مقاله	12,745,943

	On the proportion of elements of prime order in finite symmetric groups
International Journal of Group Theory
مقاله 5، دوره 13، شماره 3، آذر 2024، صفحه 251-256 اصل مقاله (407.29 K)
نوع مقاله: 2022 CCGTA IN SOUTH FLA
شناسه دیجیتال (DOI): 10.22108/ijgt.2023.135509.1810
نویسندگان
Cheryl E. Praeger^* ¹؛ Enoch Suleiman²
¹Centre for the Mathematics of Symmetry and Computation, University of Western Australia, 35 Stirling Highway, Perth 6009, Australia
²Department of Mathematics, Federal University Gashua, Yobe State, Nigeria
چکیده
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 $p\leq n<2n$.
کلیدواژه‌ها
Finite symmetric groups؛ element proportions؛ elements of prime order

مراجع
[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.
آمار تعداد مشاهده مقاله: 522 تعداد دریافت فایل اصل مقاله: 569

سامانه مدیریت نشریات علمی. طراحی و پیاده سازی از سیناوب

پیوندهای مفید

اخبار و اعلانات

آمار

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