Le groupe $H_8\times C_2$ est le plus petit groupe pour lequel existent des modules stablement libres non libres. On montre que toutes les classes d’isomorphisme de tels modules peuvent être représentées une infinité de fois par des anneaux d’entiers. On applique un travail de classification de Swan, pour cela on doit construire explicitement des bases normales d’entiers d’extensions à groupe $H_8$; cela se fait en liant un critère de Martinet avec une construction de Witt.

Let $H_8$ be the unique noncommutative and noncocommutative eight dimensional semi-simple Hopf algebra. We first construct a weak Hopf algebra $\widetilde{H_8}$ based on $H_8$, then we investigate the structure of the representation ring of $\widetilde{H_8}$. Finally, we prove that the automorphism group of $r\left(\widetilde{H_8}\right)$ is just isomorphic to $D_6$, where $D_6$ is the dihedral group with order 12.

If $G$ is a non-cyclic finite group, non-isomorphic $G$-sets $X,Y$ may give rise to isomorphic permutation representations $ℂ\left[X\right]\cong ℂ\left[Y\right]$. Equivalently, the map from the Burnside ring to the rational representation ring of $G$ has a kernel. Its elements are called Brauer relations, and the purpose of this paper is to classify them in all finite groups, extending the Tornehave–Bouc classification in the case of $p$-groups.

Let $G$ be a finite group with a Sylow 2-subgroup $P$ which is either quaternion or semi-dihedral. Let $k$ be an algebraically closed field of characteristic 2. We prove the existence of exotic endotrivial $kG$-modules, whose restrictions to $P$ are isomorphic to the direct sum of the known exotic endotrivial $kP$-modules and some projective modules. This provides a description of the group $T\left(G\right)$ of endotrivial $kG$-modules.

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 $2n$.