| Deformation theory has to do with the behavior of mathematical objects, such as group representations, under small perturbations. This theory is useful in both pure and applied mathematics and has led to the solution of many long-standing problems. In particular, in number theory, Wiles and Taylor used universal deformation rings of Galois representations in the proof of Fermat's Last Theorem. The main motivation for determining universal deformation rings for finite groups is to test or verify conjectures about the ring structure of universal deformation rings for arbitrary Galois groups. In this thesis, I study the universal deformation rings of certain 2-modular representations of two finite groups for which the representation theory is well understood and whose 2-modular blocks have been completely classified by Erdmann using quivers and relations.;More precisely, consider the symmetric group S 5 which has dihedral Sylow 2-subgroups of order 8 and its non-trivial double cover S5 whose Sylow 2-subgroups are generalized quaternion groups of order 16. Let k be an algebraically closed field of characteristic 2 and let B 0(kS5) be the principal block of kS5. In this thesis, I first classify all B 0(kS5)-modules with stable endomorphism ring isomorphic to k that also have stable endomorphism ring isomorphic to k when viewed as kS 5-modules by inflation, and then I determine the universal deformation ring modulo 2 and the universal deformation ring for each of these modules. |
