We investigate simple endotrivial modules of finite quasi-simple groups and classify them in several important cases. This is motivated by a recent result of Robinson [41] showing that simple endotrivial modules of most groups come from quasi-simple groups.
Given a dihedral 2-group P of order at least 8, we classify the splendid Morita equivalence classes of principal 2-blocks with defect groups isomorphic to P . To this end we construct explicit stable equivalences of Morita type induced by specific Scott modules using Brauer indecomposability and gluing methods; we then determine when these stable equivalences are actually Morita equivalences, and hence automatically splendid Morita equivalences. Finally, we compute the generalised decomposition numbers in each case.
a b s t r a c tThe aim of this paper is to construct an equivalent of the Dade group of a p-group for an arbitrary finite group G, whose elements are equivalence classes of endo-p-permutation modules. To achieve this goal we use the theory of relative projectivity with respect to a module and that of relative endotrivial modules.
In this paper we use projectivity relative to kG-modules to define groups of relatively endotrivial modules, which are obtained by replacing the notion of projectivity with that of relative projectivity in the definition of ordinary endotrivial modules. To achieve this goal we develop the theory of projectivity relative to modules with respect to standard group operations such as induction, restriction and inflation. As a particular example, we show how these groups can generalise the Dade group. Finally, for finite groups having a cyclic Sylow p-subgroup, we determine all the different subcategories of relatively projective modules and, using the structure of the group T (G) of endotrivial modules described in Mazza and Thévenaz (2007) [1], the structure of all the different groups of relatively endotrivial modules.
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