Solving class equation xd = β in an alternating group for all n ∈ θ &[formula omitted

Abstract

In this paper we find out the solutions to the class equation xd=β in the alternating group An for each β∈Hn∩Cα and n∈θ={1,2,5,6,10,14}, where β ranges over the conjugacy class A(β) in An and d is a positive integer number, Hn={Cα of Sn∣n>1, with all parts αk of α different and odd}, Cα is conjugacy class of Sn and form each conjugacy class Cα depends on the cycle partition α of its elements. In another direction, for any permutation λ in the symmetric group Sn, if λ∈Cα and λ∉Hn∩Cα, then Cα does not split into the two classes Cα± of An. Moreover, in the present research, the number of solutions is determined and this current work is supported by examples.