It is proved that the p-Splitting Group Basic Problem for arbitrary abelian groups and the Splitting Group Basic Problem for p-mixed  groups are, in fact, equivalent in the commutative group algebra of characteristic p. In addition, a new scheme of obtaining the complete set of invariants for such a group algebra of a p-splitting group is given. In particular, as application, the full system of invariants for a group algebra when the group is p-splitting Warfield is established as well. This supplies own recent results in (Bol. Soc. Mat. Maxicana, 2004) and (Acta Math. Sinica, 2005).