Isomorphisms Of Character Rings Over Rational Number Field

Let G be a finite abelian group andΓ_G be the Galois group over Q of the extension of Q by adding a primitive |G|-th unity root. We define an action ofΓ_G on the group algebra ZG, where Z is the ri(?)g of rational integral numbers. If the character ring of G over the Q is isomorphic to the character ring of another finite abelian group H over the Q, and CA = AC', in which the definition of A, C, C' can be found on page seven and page seventeen, then we prove (ZG)^Γ_G≌(ZH)^Γ_H .