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p-Divisibility of the Number of Solutions of in a Symmetric Group
H. Ishihara1, H. Ochiai2, Y. Takegahara3, and T. Yoshida4
1Kumamoto National College of Technology, Kumamoto 861-1102, Japan
ishihara@ge.knct.ac.jp
22Department of Mathematics, Tokyo Institute of Technology, Meguro-ku 152-8551, Japan
ochiai@math.titech.ac.jp
3Muroran Institute of Technology, Muroran 050-8585, Japan
yugen@mmm.muroran-it.ac.jp
4Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
yoshidat@math.sci.hokudai.ac.jp
Annals of Combinatorics 5 (2) p.197-210 June, 2001
AMS Subject Classification: 11B39, 11B50, 12J25, 20B30, 30G06
Abstract:
For a prime and for the number of solutions of in the symmetric group on letters, , and especially, provided . Let be an integer with . If , then, for each positive integer , . Assume that . If , then ; otherwise, there exists a -adic integer such that .
Keywords: Frobenius theorem, -adic, symmetric group

References:

1.  S. Bosch, U. G邦ntzer, and R. Remmert, Non-Archimedian Analysis, Grund. 261, Springer- Verlag, New York, 1984.

2.  W. Burnside, Theory of Groups of Finite Order, Dover, New York, 1955.

3.  S. Chowla, I.N. Herstein, and W.K. Moore, On recursions connected with symmetric groups I, Canad. J. Math. 3 (1951) 328每334.

4.  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每31.