It has been an open question
for some time whether for every noncommutative division ring D finite
dimensional over its center, the multiplicative group D^* = D \ {0} has
a maximal proper subgroup. We prove that the answer is yes, except
possibly for some unusual division algebras that probably don't exist.
The difficult case in the proof is that of a quaternion division
algebra over a Euclidean field, which we handle by a geometric
argument. This is joint work with R. Hazrat.