On permutations of order dividing a given integer
Alice C. Niemeyer
and Cheryl E. Praeger
University of Western Australia School of Mathematics and Statistics Nedlands WA 6907 Australia
DOI: 10.1007/s10801-007-0056-5
Abstract
We give a detailed analysis of the proportion of elements in the symmetric group on n points whose order divides m, for n sufficiently large and m\geq n with m= O( n).
Pages: 125–142
Keywords: keywords symmetric group; proportions
Full Text: PDF
References
1. Beals, R., Leedham-Green, C. R., Niemeyer, A. C., Praeger, C. E., & Seress, Á. (2003). A black-box group algorithm for recognizing finite symmetric and alternating groups. I. Trans. Am. Math. Soc. (electronic), 355(5), 2097-2113.
2. Bouwer, I. Z., & Chernoff, W. W. (1985). Solutions to xr = αin the symmetric group. In Tenth British Combinatorial conference (Glasgow, 1985). Ars Comb. (A), 20, 83-88.
3. Chowla, S., Herstein, I. N., & Scott, W. R. (1952). The solutions of xd = 1 in symmetric groups. Norske Vid. Selsk., 25, 29-31.
4. Erd\Acute\Acute os, P., & Turán, P. (1965). On some problems of a statistical group-theory. I. Wahrsch. Verw. Geb., 4, 175-186.
5. Erd\Acute\Acute os, P., & Turán, P. (1967). On some problems of a statistical group-theory. III. Acta Math. Acad. Sci. Hung., 18, 309-320.
6. Gao, L., & Zha, J. G. (1987). Solving the equation xn = σin the symmetric group Sm. J. Math. (Wuhan), 7(2), 173-176.
7. Landau, E. (1909). Handbuch der Lehre von der Verteilung der Primzahlen. Leipzig: Teubner.
8. Mineev, M. P., & Pavlov, A. I. (1976). An equation in permutations. Trudy Mat. Inst. Steklov., 142(270), 182-194.
9. Mineev, M. P., & Pavlov, A. I. (1976). The number of permutations of a special form, Mat. Sbornik (N.S.), 99(141)(3), 468-476, 480.
10. Moser, L., & Wyman, M. (1955). On solutions of xd = 1 in symmetric groups. Can. J. Math., 7, 159-168.
11. Moser, L., & Wyman, M. (1956). Asymptotic expansions. Can. J. Math., 8, 225-233. J Algebr Comb (2007) 26: 125-142
12. Niemeyer, A. C., & Praeger, C. E. (2005). On the proportion of permutations of order a multiple of the degree. Preprint.
13. Niemeyer, A. C., & Praeger, C. E. (2006). On the frequency of permutations containing a long cycle. J. Algebra, 300, 289-304.
14. Niven, I., Zuckerman, H. S., & Montgomery, H. L. (1991). An introduction to the theory of numbers (5th ed.). New York: Wiley.
15. Volynets, L. M. (1986). The number of solutions of the equation xs = e in a symmetric group. Mat. Zametki, 40, 155-160, 286.
16. Warlimont, R. (1978). Über die Anzahl der Lösungen von xn = 1 in der symmetrischen Gruppe Sn. Arch. Math. (Basel), 30(6), 591-594.
17. Wilf, H. S. (1986). The asymptotics of eP (z) and the number of elements of each order in Sn. Bull. Am. Math. Soc. (N.S.), 15(2), 228-232.
2. Bouwer, I. Z., & Chernoff, W. W. (1985). Solutions to xr = αin the symmetric group. In Tenth British Combinatorial conference (Glasgow, 1985). Ars Comb. (A), 20, 83-88.
3. Chowla, S., Herstein, I. N., & Scott, W. R. (1952). The solutions of xd = 1 in symmetric groups. Norske Vid. Selsk., 25, 29-31.
4. Erd\Acute\Acute os, P., & Turán, P. (1965). On some problems of a statistical group-theory. I. Wahrsch. Verw. Geb., 4, 175-186.
5. Erd\Acute\Acute os, P., & Turán, P. (1967). On some problems of a statistical group-theory. III. Acta Math. Acad. Sci. Hung., 18, 309-320.
6. Gao, L., & Zha, J. G. (1987). Solving the equation xn = σin the symmetric group Sm. J. Math. (Wuhan), 7(2), 173-176.
7. Landau, E. (1909). Handbuch der Lehre von der Verteilung der Primzahlen. Leipzig: Teubner.
8. Mineev, M. P., & Pavlov, A. I. (1976). An equation in permutations. Trudy Mat. Inst. Steklov., 142(270), 182-194.
9. Mineev, M. P., & Pavlov, A. I. (1976). The number of permutations of a special form, Mat. Sbornik (N.S.), 99(141)(3), 468-476, 480.
10. Moser, L., & Wyman, M. (1955). On solutions of xd = 1 in symmetric groups. Can. J. Math., 7, 159-168.
11. Moser, L., & Wyman, M. (1956). Asymptotic expansions. Can. J. Math., 8, 225-233. J Algebr Comb (2007) 26: 125-142
12. Niemeyer, A. C., & Praeger, C. E. (2005). On the proportion of permutations of order a multiple of the degree. Preprint.
13. Niemeyer, A. C., & Praeger, C. E. (2006). On the frequency of permutations containing a long cycle. J. Algebra, 300, 289-304.
14. Niven, I., Zuckerman, H. S., & Montgomery, H. L. (1991). An introduction to the theory of numbers (5th ed.). New York: Wiley.
15. Volynets, L. M. (1986). The number of solutions of the equation xs = e in a symmetric group. Mat. Zametki, 40, 155-160, 286.
16. Warlimont, R. (1978). Über die Anzahl der Lösungen von xn = 1 in der symmetrischen Gruppe Sn. Arch. Math. (Basel), 30(6), 591-594.
17. Wilf, H. S. (1986). The asymptotics of eP (z) and the number of elements of each order in Sn. Bull. Am. Math. Soc. (N.S.), 15(2), 228-232.