Some properties of interior G-algebras
Let G be a finite group. Let R be a complete discrete valuation ring with residue field k of characteristic p, which is a splitting field for G or let \(R=k\). The author calls a pair (A,\(\rho)\), where A is an R-algebra, free as R-module, and \(\rho\) is an R-algebra homomorphism from RG onto A, epimorphic interior G-algebra. It is called local, if A is indecomposable as an R(G\(\times G)\)-module (with the obvious \(G\times G\)-action). These pairs are special interior G-algebras, in the sense of Puig, and so defect groups are defined. After a section with basic properties of epimorphic local interior G-algebras and their defect groups the author defines an invariant related to the defect groups of these G-algebras. More precisely, as a consequence of the author's result on the vertex of A as R(G\(\times G)\)-module [J. Algebra 105, 344-350 (1987; Zbl 0604.20007)] the following holds. Let (A,\(\rho)\) be an epimorphic local interior G-algebra with fixed defect group D. Then there exists \(vtx_{G\times G}A\) which is contained in \(D\times D\) and contains the diagonal of D. The intersection of \(vtx_{G\times G}A\) with the first factor of \(G\times G\) is a normal subgroup of D and this invariant is unique up to conjugation in \(N_ G(D).\) Most of the paper is devoted to the study of this invariant. The reduction by a normal subgroup and the case when the invariant is cyclic is considered. The author gives a proof of a theorem of Michler about the embedding of defect groups into blocks using interior G-algebras (here the invariant is 1) and considers, finally, epimorphic local interior G- algebras exemplarily in the case when G is a p-group.
scientific article; zbMATH DE number 4027734