Solving class equation xd =b in an alternating group for all n 2 h & b 2 Hn \ Ca

Abstract

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