Generalization of the basis theorem for alternating groups

There were defined in the paper of Robert Shwartz [3] OGS for symmetric groups, as an interesting generalization of the basis of finite abelian groups. The definition of OGS states that that every element of a symmetric group has a unique presentation as a product of some powers of the OGS, un a specific given order. The same paper has demonstrated a strong connection between the OGS and the standard Coxeter presentation of the symmetric group. The OGS presentation helps us to find the Coxeter length and the descent set of an arbitrary element of the symmetric group. Therefore, it motivates us to generalize the OGS for the alternating subgroup of the symmetric group, which we define in this paper. We generalize also the exchange laws for the alternating subgroup, and we will show some interesting properties of it.