Let k be an algebraically closed field of characteristic 2, and let W be the
ring of infinite Witt vectors over k. Suppose G is a finite group and B is a
block of kG with a dihedral defect group D such that there are precisely two
isomorphism classes of simple B-modules. We determine the universal deformation
ring R(G,V) for every finitely generated kG-module V which belongs to B and
whose stable endomorphism ring is isomorphic to k.

Let $k$ be an algebraically closed field of characteristic 2, let $G$ be a
finite group with dihedral Sylow 2-subgroups, and let $B$ be the principal
block of $kG$. Assume that there are precisely two isomorphism classes of
simple $B$-modules. The description by Erdmann of the quiver and relations of
the basic algebra of $B$ is usually only given up to a certain parameter $c$
which is either 0 or 1. In this article, we show that $c=0$ if there exists a
central extension $\hat{G}$ of $G$ by a group of order 2 such that the Sylow
2-subgroups of $\hat{G}$ are generalized quaternion.

We determine the universal deformation ring R(G,V) of certain mod 2
representations V of a finite group G whose Sylow 2-subgroups are isomorphic to
a generalized quaternion group D. We show that for these V, a question raised
by the author and Chinburg concerning the relation of R(G,V) to D has an
affirmative answer. We also show that R(G,V) is a complete intersection even
though R(G/N,V) need not be for certain normal subgroups N of G which act
trivially on V.