Solving the Class Equation x^{d} = β in an Alternating Group for Each β ∈ C^{α} ∩ H_{n}^{c} and n > 1

Affiliation(s)

Department of Mathematics, College of Science, University of Basra, Basra, Iraq.

School of Mathematical Sciences, Universiti Sains Malaysia, Penang, Malaysia.

Department of Mathematics, College of Science, University of Basra, Basra, Iraq.

School of Mathematical Sciences, Universiti Sains Malaysia, Penang, Malaysia.

ABSTRACT

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 .

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 .

Cite this paper

S. Mahmood and A. Rajah, "Solving the Class Equation x^{d} = β in an Alternating Group for Each β ∈ C^{α} ∩ H_{n}^{c} and n > 1," *Advances in Linear Algebra & Matrix Theory*, Vol. 2 No. 2, 2012, pp. 13-19. doi: 10.4236/alamt.2012.22002.

S. Mahmood and A. Rajah, "Solving the Class Equation x

References

[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

[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