Indecomposable modules, relative projectivity and radical subgroups in finite groups

Abstract

In the various blocks of a finite group G, irreducible characters sit with the indecomposable modules which afford them and such indecomposable modules in those blocks have got vertices and sources and in fact, every p-subgroup of G is a vertex of some indecomposable FG-module. For any finite group G and a field F of characteristic p, where p is a prime that divides the order |G| of G, every indecomposable FG-module possesses a vertex and a source. Furthermore for a finite group G, kernels of the irreducible FG-modules, vertices of the irreducible FG-modules and defect groups of blocks of G all contain Op(G). Furthermore, the kernels of blocks of G are normal p′-subgroups of G which are contained in Op′(G), where Op′(G) is the kernel of the principal block of G. The kernels of blocks of G are related to the kernels of the indecomposable FG-modules in those blocks. The object in this paper is to study characteristics and/or properties of vertices of indecomposable FG-modules in relation to irreducible ordinary characters that they afford and sit with in blocks of G and furthermore study characteristics and/or properties that exist between kernels of irreducible FG-modules, vertices of irreducible FG-modules and defect groups of blocks of G and even establish as to when and how any and/or all of these would (if at all possible) coincide.