Applications on new category of the symmetric groups

In this work, for any symmetric group we determine all the integer values taken by group determinant when the matrix entries are integers. The new types of finite groups(M (Sn), ⊗) and (M (An), ⊗) are given and studied. They are called Matrix groups generated by Sn and An, respectively. These new types of groups are discussed their prosperities and this work is supported by some examples. We show that (M (An), ⊗) is a subgroup of (M (Sn), ⊗) and M (gk) = M (gh), for any two permutations are conjugated (gk ≅ gh) in Sn. However, It is not necessary M (gk) ≠ M (gh) for any two permutations gk, gh are not conjugate in An.