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

We describe the structure of the integral group ring imageG, 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 imageG is of finite representation type, we deduce the structure of all genera of imageG-lattices. Our principal applications are the following. (i) We determine which groups G have the property that each imageG-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 imageG-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 imageG determines the group G up to isomorphism.