Universal deformation rings, endo-trivial modules, and semidihedral and generalized quaternion 2-groups

Abstract

Let k be a field of characteristic p>0, and let W be a complete discrete valuation ring of characteristic 0 that has k as its residue field. Suppose G is a finite group and Gab,p is its maximal abelian p-quotient group. We prove that every endo-trivial kG-module V has a universal deformation ring that is isomorphic to the group ring WGab,p. In particular, this gives a positive answer to a question raised by Bleher and Chinburg for all endo-trivial modules. Moreover, we show that the universal deformation of V over WGab,p is uniquely determined by any lift of V over W. In the case when p=2 and G is a 2-group that is either semidihedral or (generalized) quaternion, we give an explicit description of the universal deformation of every indecomposable endo-trivial kG-module V.