Special Quasi-triads and Integral Group Rings of Finite Representation Type, II

Abstract

We describe the structure of the integral group ring ZG, when G is a group with cyclic Sylow subgroups, as a subdirect sum of hereditary orders in cyclic crossed product algebras. In the case where ZG is of finite representation type, we deduce the structure of all genera of ZG-lattices. Our principal applications are the following. (i) We determine which groups G have the property that each ZG-lattice is isomorphic to a direct sum of right ideals. (ii) We determine which groups G with cyclic Sylow subgroups have the property that each ZG-lattice has a unique number of indecomposable summands. (iii) We show that, for groups G with cyclic Sylow subgroups, the ring structure of the rational group algebra QG determines the group G up to isomorphism.