Solving the Class Equation xd = β in an Alternating Group for Each β ∈ Cα ∩ Hnc and n > 1

The main purpose of this paper is to solve the class equation in an alternating group, (i.e. find the solutions set ) and find the number of these solutions where ranges over the conjugacy class in and d is a positive integer. In this paper we solve the class equation in where , for all . is the complement set of where { of , with all parts of are different and odd}. is conjugacy class of and form class depends on the cycle type of its elements If and , then splits into the two classes of .

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

S. Mahmood and A. Rajah, "Solving the Class Equation xd = β in an Alternating Group for Each β ∈ Cα ∩ Hnc and n > 1," Advances in Linear Algebra & Matrix Theory, Vol. 2 No. 2, 2012, pp. 13-19. doi: 10.4236/alamt.2012.22002.

 [1] H. Ishihara, H. Ochiai, Y. Takegahara and T. Yoshida, “p-Divisibility of the Number of Solutions of xp = 1 in a Sym-metric Group,” Annals of Combinatorics, Vol. 5, No. 2, 2001, pp. 197-210. doi:10.1007/PL00001300 [2] N. Chigira, “The Solutions of xd = 1 in Finite Groups,” Journal of Algebra, Vol. 180, No. 3, 1996, pp. 653-661. doi:10.1006/jabr.1996.0086 [3] R. Brauer, “On a Theorem of Frobenius,” American Ma- thematical Monthly, Vol. 76, No. 1, 1969, pp. 12-15. doi:10.2307/2316779 [4] Y. G. Berkovich, “On the Number of Elements of Given Order in Finite p-Group,” Israel Journal of Mathematics, Vol. 73, No. 1, 1991, pp. 107-112. doi:10.1007/BF02773429 [5] T. Ceccherini-Silberstein, F. Scarabotti and F. Tolli, “Rep- resentation Theory of the Sym-metric Groups,” Cambridge University Press, New York, 2010. [6] D. Zeindler, “Permutation Matrices and the Mo-ments of Their Characteristic Polynomial,” Electronic Journal of Probability, Vol. 15, No. 34, 2010, pp. 1092-1118. [7] J. J. Rotman, “An Introduction to the Theory of Groups,” 4th Edition, Springer-Verlag, New York, 1995. [8] G. D. James and A. Kerber, “The Representation Theory of the Symmetric Group,” Addison-Wesley Publishing, Boston, 1984. [9] S. A. Taban, “Equations in Symmetric Groups,” Ph.D. Thesis, University of Basra, Basra, 2007. [10] S. Mahmood and A. Rajah, “Solving the Class Equation xd = β in an Alternating Group for each and ,” Journal of the Association of Arab Universities for Basic and Applied Sciences, Vol. 10, No. 1, 2011, pp. 42-50. doi:10.1016/j.jaubas.2011.06.006