Product of conjugacy classes of the alternating group An

الملخص EN

For a nonempty subset X of a group G and a positive integer m, the product of X, denoted by Xm, is the set Xm =         m I i i x x X 1 : That is, Xm is the subset of G formed by considering all possible ordered products of m elements form X.

In the symmetric group Sn, the class Cn (n odd positive integer) split into two conjugacy classes in an denoted Cn + and Cn -.

C+ and C- were used for these two parts of Cn.

This work we prove that for some odd n, the class C of 5- cycle in Sn has the property that 2 n3 C = An n 7 and C+ has the property that each element of C+ is conjugate to its inverse, the square of each element of it is the element of C-, these results were used to prove that C + C- = An exceptional of I (I the identity conjugacy class), when n = 5 + 4k , k > = 0.