Class groups of integral group rings

Abstract:

Let $\Lambda$ be an $R$-order in a semisimple finite dimensional $K$-algebra, where $K$ is an algebraic number field, and $R$ is the ring of algebraic integers of $K$. Denote by $C(\Lambda )$ the reduced class group of the category of locally free left $\Lambda$-lattices. Choose $\Lambda = ZG$, the integral group ring of a finite group $G$, and let $\Lambda ’$ be a maximal $Z$-order in $QG$ containing $\Lambda$. There is an epimorphism $C(\Lambda ) \to C(\Lambda ’)$, given by $M \to \Lambda ’{ \otimes _\Lambda }M$, for $M$ a locally free $\Lambda$-lattice. Let $D(\Lambda )$ be the kernel of this epimorphism; the groups $D(\Lambda ),C(\Lambda )$ and $C(\Lambda ’)$ are all finite. Our main theorem is that $D(ZG)$ is a $p$-group whenever $G$ is a $p$-group. This generalizes Fröhlich’s result for the case where $G$ is an abelian $p$-group. Our proof uses some facts about the center $F$ of $QG$, as well as information about reduced norms. We also calculate $D(ZG)$ explicitly for $G$ cyclic of order $2p$, dihedral of order $2p$, or the quaternion group. In these cases, the ring $ZG$ can be conveniently described by a pullback diagram.