Normal subgroups and support τ-tilting modules

Let G be a finite group, G a normal subgroup of G and k an algebraically closed field of characteristic p > 0. The first main result in this paper is to show that support τ-tilting kG̃-modules with some properties are support τ-tilting modules as kG-modules, too. As the second main result, we give equivalent conditions for support τ-tilting kG̃-modules to satisfy the above properties, and show that the set of the support τ-tilting kG̃-modules with the properties is isomorphic to the set of G̃-invariant support τ-tilting kG-modules as posets. As an application, we show that the set of G̃-invariant support τ-tilting kG-modules is isomorphic to the set of support τ-tilting kG̃-modules in the case that the index of G in G̃ is a p-power. As a further application, we give a feature of vertices of indecomposable τ-rigid kG̃-modules. Finally, we give block versions of the above results.