Special quasi-triads and integral group rings of finite representation type. II
Let \(G\) be a finite group with cyclic Sylow \(p\)-subgroups for all primes \(p\) dividing \(| G|\). In the paper under review, first the rational group algebra \(\mathbb{Q} G\) is described as a direct sum of cyclic crossed product algebras. As a consequence, it is shown that, for such \(G\), the isomorphism type of \(\mathbb{Q} G\) determines that of \(G\). Then, for every prime \(p\), the localization \(\mathbb{Z}_{(p)}G\) and the completion \(\mathbb{Z}_ pG\) are described as direct sums of iterated pullbacks. When the Sylow \(p\)-subgroups of \(G\) have order dividing \(p^ 2\), the indecomposable \(\mathbb{Z}_ pG\)-lattices are determined using the results of part I [see the preceding review Zbl 0790.20013)] of this paper. This leads to a method of constructing the genera of all indecomposable \(\mathbb{Z} G\)-lattices. As an application, the authors are able to classify all finite groups \(G\) having the property that every indecomposable \(\mathbb{Z} G\)-lattice is isomorphic to a right ideal. Moreover, they determine those finite groups \(G\) with cyclic Sylow subgroups having the property that every \(\mathbb{Z} G\)-lattice has a unique number of indecomposable direct summands.
scientific article; zbMATH DE number 480476