Representations of finite groups over number rings

Let $R'$ be the ring of integers of a finite extension $F'$ of the field of rational $p$-adic numbers $Q_p$, and let $G$ be a finite group. All groups $G$ and fields $F'$ are found such that the number of indecomposable representations of $G$ over $R'$ is finite. In addition, we investigate the problem of complete reducibility of a matrix $R'$-representation of an abelian $p$-group, all of whose irreducible components are $F'$-equivalent.