Commutative group rings of finite representation type

Abstract

Let Z p denote the localization (but not the completion) of the integers at the prime p. Then, for finite groups G, the group ring Z p G has finite representation type if and only if the p-Sylow subgroups of G are cyclic of order at most p 2. In this paper, we determine the possible ranks of indecomposable Z p G-lattices for finite abelian groups G for which Z p G has finite representation type. In particular, for such groups G, we show that every indecomposable Z p G-lattice can be embedded as a sublattice of Z p G (4), but not, in general, as a sublattice of Z p G (3).