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 $G^{\mathrm{ab},p}$ 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 $WG^{\mathrm{ab},p}$. 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 $WG^{\mathrm{ab},p}$ is uniquely determined by any lift of $V$ over $W$. In the case when $p=2$ and $G=\mathrm{D}$ 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 $k\mathrm{D}$-module $V$.