Solving class equation xd = β in an alternating group for all n ∈ θ & β∈Hn∩Cαβ∈Hn∩Cα

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.