Solving the class equation xd = β in an alternating group for each β ∈ H ∩ Cα and n ∉ θ

Abstract

In this paper we find the solutions to the class equation xd = β in the alternating group An (i.e. find the solutions set X = {x ∈ An∣xd ∈ A(β)}) and find the number of these solutions ∣X∣ for each β ∈ H ∩ Cα and n ∉ θ, where H = {Cα of Sn∣n > 1, with all parts αk of α different and odd}. Cα is a conjugacy class of Sn and forms classes, where Cα depends on the cycle partition α of its elements. If (14 > n ∉ θ ∪ {9, 11, 13}) and β ∈ H ∩ Cα in An, then Fn contains Cα, where Fn = {Cα of Sn∣ the number of parts αk of α with the property αk ≡ 3 (mod 4) is odd}. In this work we introduce several theorems to solve the class equation xd = β in the alternating group An where β ∈ H ∩ Cα and n ∉ θ and we find the number of the solutions for n to be: (i) 14 > n ∉ θ, (ii) 14 > n ∉ θ and (n + 1) ∉ θ, (iii) 14 > n ∉ θ and Cα ≠ [1, 3, 7], (iv) n = 9, 11, 13, (v) n > 14.