Universal deformation rings and semidihedral 2-groups

The main objective of deformation theory is to study the behavior of mathematical objects, such as modules or group representations, under perturbations. This theory is useful in both pure and applied mathematics and has led to the solution of many long-standing problems. For example, in number theory, universal deformation rings of Galois representations played an important role in the proof of Fermat’s Last Theorem by Wiles and Taylor.

In this thesis, we consider the case when SD_n is a semidihedral 2-group of order 2ⁿ⁺¹ for n ≥ 3 and k is an algebraically closed field of characteristic 2. The indecomposable kSD_n-modules have been completely described by Bondarenko and Drozd, and Crawley-Boevey. We concentrate on so-called endo-trivial kSD_n-modules, which possess a well-defined universal deformation ring by work of Bleher and Chinburg. Using the classification of Carlson and Thevenaz of all endo-trivial kSD_n-modules, we show that the universal deformation ring of every endo-trivial kSD_n-module is isomorphic to the group ring W [ℤ/2 x ℤ/2], where W = W (k) is the ring of infinite Witt vectors over k.