Subgroup complexes of finite groups, and their representation and geometry

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In order to study the structure theory of finite groups, it is quite important to investigate that how various subgroups of the group relate or intersect each other. Then we consider an abstract simplicial complex defined by inclusion chains of subgroups as simplices ; which is called "subgroup complex". We obtained some results on the Lefschetz module, the generalized Burnside ring, or the homotopy invarinat associated to subgroup complexes.