Solving the class equation xd = β in an alternating group for each β ε H ∩ Ca and n ε Q

Abstract EN

In this paper we find the solutions to the class equation xd =b in the alternating group An (i.

e.

find the solutions set X = {x 2 AnOExd 2 A(b) } ) and find the number of these solutions OEXOE for each b 2 H \ Ca and n R h, where H = {Ca of SnOEn > 1, with all parts ak of a different and odd}.

Ca is conjugacy class of Sn and form class Ca depends on the cycle partition a of its elements. If (14 >n R h [ {9, 11, 13}) and b 2 H \ Ca in An, then Fn contains Ca, where Fn = {Ca of SnOE the number of parts ak of a with the property ak ” 3 (mod 4) is odd}.

In this work we introduce several theorems to solve the class equation xd= b in the alternating group An where b 2 H \ Ca and n R h and we find the number of the solutions for n to be : (i) 14 > n R h, (ii) 14 > n R h and (n+ 1) R h, (iii) 14 > n R h and Ca „ [1, 3, 7], (iv) n = 9, 11, 13, (v) n >14.