The main results of this paper may be loosely stated as follows.Theorem.— Let $N$ and $N^{\text{'}}$ be sums of Galois algebras with group $\Gamma$ over algebraic number fields. Suppose that $N$ and $N^{\text{'}}$ have the same dimension $\mathbb{Q}$ and that they are identical at their wildly ramified primes. Then (writing ${𝒪}_N$ for the maximal order in $N$)$${𝒪}_N\oplus {𝒪}_N\oplus \mathbb{Z}\Gamma {\cong }_{\mathbb{Z}\Gamma }\oplus {𝒪}_N^{\text{'}}\oplus {𝒪}_{N^{\text{'}}}\oplus \mathbb{Z}\Gamma .$$In many cases ${𝒪}_N{\cong }_{\mathbb{Z}\Gamma }{𝒪}_{N^{\text{'}}}.$The role played by the root numbers of $N$ and $N^{\text{'}}$ at the symplectic characters of $\Gamma$ in determining the relationship between the $\mathbb{Z}\Gamma$-modules ${𝒪}_N$ and ${𝒪}_{N^{\text{'}}}$ is described. The theorem includes...

Let $kG$ be a group algebra, and $D\left(kG\right)$ its quantum double. We first prove that the structure of the Grothendieck ring of $D\left(kG\right)$ can be induced from the Grothendieck ring of centralizers of representatives of conjugate classes of $G$. As a special case, we then give an application to the group algebra $k{D}_n$, where $k$ is a field of characteristic $2$ and $D_n$ is a dihedral group of order $2n$.